COEXISTENCE AND ASYMPTOTIC STABILITY INSTAGE-STRUCTURED PREDATOR-PREY MODELS

Michael Cowen

A Thesis Submitted to theUniversity of North Carolina Wilmington in Partial Fulfillment

of the Requirements for the Degree ofMaster of Arts

Department of Mathematics and Statistics

University of North Carolina Wilmington

2013

Approved by

Advisory Committee

Xin Lu Michael Freeze

Wei Feng

Chair

Accepted by

Dean, Graduate School

This thesis has been prepared in the style and format

Consistent with the journal

American Mathematical Monthly.

ii

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 A NON-STAGE STRUCTURED MODEL . . . . . . . . . . . . . . . 12

4 THE ULTIMATE BOUNDS FOR THE POPULATIONS . . . . . . . 22

4.1 Extinction of the predator by death rate and prey handling time 22

4.2 Extinction of the total population by birth and death rate of

the adult prey . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 EQUILIBRIUM POINTS WITH EXTINCTION OF PREDATOR . . 24

5.1 Equilibrium for extinction of predator and prey . . . . . . . . 24

5.2 Equilibrium for extinction of predator only . . . . . . . . . . . 28

6 COEXISTENCE WHILE ONLY ONE STAGE OF THE PREY IS

VULNERABLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.1 Juveniles are vulnerable . . . . . . . . . . . . . . . . . . . . . 34

6.2 Adults are vulnerable . . . . . . . . . . . . . . . . . . . . . . . 43

7 COEXISTENCEWHILE JUVENILE ANDADULT PREY ARE BOTH

VULNERABLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

8 DISCUSSION/CONCLUSION . . . . . . . . . . . . . . . . . . . . . . 66

iii

ABSTRACT

In this paper we analyze the effects of a stage-structured predator-prey system

where the prey has two stages, juvenile and adult. Three different models (where

the juvenile or adult prey populations are vulnerable) are studied to evaluate the

impacts of this structure to the stability of the system and coexistence of the species.

We assess how various ecological parameters, including predator mortality rate and

handling times on prey, prey growth rate and death rate, as well as prey capture

rate and nutritional values in two stages, affect the existence and stability of all

possible equilibria in each of the models. The main focus of this paper is to find

general conditions to ensure the presence and asymptotic stability of the equilibrium

point where both the predator and prey can co-exist. Through specific examples, we

demonstrate the stability of the co-existence equilibrium and the dynamics in each

system.

iv

ACKNOWLEDGMENTS

I would like to thank those who have helped me with this thesis and who have

helped me throughout my graduate career. To Dr. Feng: Thank you for helping

me complete this thesis. Thank you for encouraging me to continue to work on this

model and for encouraging me to present at the conference over the summer. I would

also like to thank my committee members Dr. Lu and Dr. Freeze. Thank you for

your help and your suggestions with this thesis. To Dr. Freeze: I have enjoyed all of

the graduate classes that I have taken with you. You have always been very helpful

with all sorts of mathematical problems that I have asked about. Thank you also

for the insights that you have given me in regards to my work with this thesis. To

Dr. Lu: Thank you for getting me involved with the AIMS conference. It was a

great experience and I am glad that I can continue to be a part of it. Thanks for all

of your patience and also thank you for the numerical simulations that you provided

for my thesis.

v

1 INTRODUCTION

Using mathematics to model the size of a population was first implemented by

Reverend Thomas Malthus in 1798 who published his findings in An Essay on the

Principle of Population. The model that he introduces, called the Malthusian growth

model has the form:

P (t) = P0ert

where

• P0 = P (0) = initial population

• r = growth rate

• t = time

After reading about Malthus’ discovery, Pierre Francois Verhulst published the

logistic growth equation in 1838 described as follows:

dN

dt= rN(1− N

K)

where

• N(t) represents the number of individuals at time t

• r is the intrinsic growth rate

• K is the carrying capacity

Over the next century, other population models began to arise; however, it was

not until 1925, where Alfred J. Lotka and Vito Volterra proposed a system of equa-

tions that modeled the interaction between two species. The model is described

as:

dx

dt= αx− βxy

dy

dt= δxy − γy

where x and y are the prey and predator population densities respectively,dx

dtand

dy

dtare the population growth rates and the parameters α, β, δ and γ represent the

interactions between the prey and predator. Since 1925, various ecological models

have been developed as a system of differential equations to study predator-prey

interactions with effects of diffusion, functional responses, time delays and stage-

structure (see, for a few examples, [4, 16, 9, 10, 11, 12, 13, 14, 19, 2, 8, 21, 24]).

It has also been realized that the changes of a certain organism has an impact on

the food chain with which it is involved. If an organism grows radically throughout

its life cycle, the amount it eats, and the amount of it which can be consumed by

its predator, if it has one, will most likely change as well. For this reason, research

regarding non-homogeneous populations have started to appear in the last forty

years [20, 18, 7, 15, 21]. Various work has been done where stage-structure is used

on predator [2, 22, 3] and prey [1, 6, 15, 21]. In 1983, Hastings [15] introduced

differential equation models with a stage-structured prey, which are similar to the

ones studied in our paper. In 2005, Abrams and Quince [1] used Hastings models to

analyze the impact of mortality on predator population size and stability in systems

with stage-structured prey, under a simplified condition where the adult prey and

juvenile prey have the same relative nutritional value (e1 = e2 = 1)to the predator.

Two of the special cases they considered are: only juvenile prey are vulnerable and

only adult prey are vulnerable, which are both extreme cases for the stage-structured

model. Although there are many examples in nature where a predator will mostly

2

consume only young prey, usually due to the dangers of attacking the larger adult

prey, there are still times even in these scenarios where the predator will attack the

adult prey. Possible explanations for this include vulnerability of the adult prey,

desperation of the predator, or teamwork of several predators. For this reason, it is

also important to consider the generalized model where both young and adult prey

are vulnerable, with various ecological parameters included.

The analysis that is given in this paper focuses on predator-prey systems, in

which the prey has two-stages (juvenile and adulthood), and the predator has one.

The system is described by the following system of equations:

dN1

dt= b2N2 − d1N1 − gN1 − s1N1PH(e1s1h1N1 + e2s2h2N2)

dN2

dt= gN1 − d2N2 − s2N2PH(e1s1h1N1 + e2s2h2N2)

dP

dt= P ((e1s1N1 + e2s2N2) ∗H(e1s1h1N1 + e2s2h2N2)−D)

(1)

where the variables, functions and parameters are defined as follows: N1 and N2

are the population densities of juvenile and adult prey, P is the population density

of the predator, b2 is the per capita birth rate of adult prey, which is presented as

a constant here since the model assumes density-independent prey growth, d1 and

d2 are parameters giving the per capita death rates of juvenile and adult prey, D

is the per capita death rate of the predator, g is the per capita transition rate of

juveniles to adults in the prey population, si is the per capita capture rate of prey

in class i by a searching predator, hi is the predator’s handling time of prey of class

i and ei represents the relative nutritional value of an individual of prey type i to

the predator. H is a function describing predator satiation defined as: H(N) = 11+N

[1].

The focus of this paper is to analyze three different models utilizing stage struc-

tures on the prey while the juveniles, or adults, or both are vulnerable to the preda-

3

tor. We find the conditions to ensure the existence and asymptotic stability of the

equilibrium points, especially the one where both the predator and prey can co-exist.

The main goal is to show, in each of these models, how the ecological parameters

(the birth rate, death rate and transition rate of the prey; the mortality rate of

the predator; the predator capture rates, handling times, and relative nutritional

values for different prey stages) affect the dynamics of the populations. For this

purpose, we will consider three possible equilibrium points: (0, 0, 0), (N1, N2, 0), and

(N1, N2, P ), which represent different ecological outcomes in the system. for the

special cases where prey is only vulnerable at one stage, we give explicit stability

conditions and further analysis on the effect of various parameters. These results

are consistent with the discussions in [1] under specific assumptions for certain pa-

rameters. For the general model where prey is vulnerable at both stages, we obtain

the coexistence equilibrium of the model through the population ratio of adult prey

to juvenile prey. This expression also enables us to further analyze the Jacobian

matrix of the system and find conditions for asymptotic stability and instability of

the coexistence equilibrium. Each of our main results is accompanied by an exam-

ple of numerical simulations, and the ecological interpretations of the mathematical

analysis are given in the discussion/conclusion section at the end of the article.

4

2 PRELIMINARIES

In physics, engineering and biology, mathematical models are frequently used to

aid in various problems that occur in these disciplines. Often, these models consist

of various equations which usually contain derivatives of an unknown function.

Definition 2.1. A differential equation is an equation that relates the value of

an unknown quantity with various orders of its derivative.

Examples of models that contain differential equations are the decay of a ra-

dioactive substance, the position of a pendulum and the population size of a certain

species. Differential equations involve rates of change dependent on various elements

such as particles, charges and time. In addition, these equations contain dependent

and independent variables.

Definition 2.2. A dependent variable is the variable in a differential equation

whose output is a result of another variable.

Definition 2.3. An independent variable is the variable in a differential equation

which affects the value of another variable in the equation.

Example 2.4. d2ydx

+ dydx

+ x = 0

In the example above, y is the dependent variable and x is the independent

variable.

Definition 2.5. An ordinary differential equation is an equation that contains

a function of one independent variable and its derivatives.

Definition 2.6. A partial differential equation is an equation that contains

unknown multivariable equations and their partial derivatives.

Example 2.7. The following examples illustrate the difference between an ODE(Ordinary

differential equation) and a PDE(Partial differential equation).

5

(a) d2ydx2 − 3 dy

dx+ 4y = 0

(b) d2udx2 +

d2udy2

= 0

In part (a), there are is only one independent variable, therefore, it is an ordinary

differential equation. The example in part (b) is a partial differential equation since

both x and y are independent variables, and the equation contains derivatives with

respect to each of these.

Definition 2.8. The order of a differential equation is the degree of the highest-

ordered derivative in the equation.

Example 2.9. The following examples are ordinary differential equations.

(a) d4ydx4 − 2x dy

dx+ 3y = 0

(b) d2ydx2 +

dydx

= 0

The equation in part (a) is fourth-order ordinary differential equation since the

highest derivative present is the fourth derivative with respect to x. The equation

in part(b) is a second-order differential equation since the highest derivative present

is the second derivative with respect to x. The model that we explore in this paper

only contains ordinary differential equations so from we will restrict ourselves to

only this type of differential equation. Within the set of ordinary equations, there

are two subsets, linear and nonlinear.

Definition 2.10. A linear differential equation has the format:

an(x)dnydxn + an−1(x)

dn−1ydxn−1 + . . .+ a1(x)

dydx

+ a0(x)y = F (x),

6

where an(x), an−1(x), ..., a0(x) and F (x) only dependent on the independent variable,

x.

Definition 2.11. An ordinary differential equation is nonlinear if it is not linear.

Example 2.12. The following examples are ordinary differential equations.

(a) d3ydx3 + 4 dy

dx+ y2 = 0

(b) d2ydx2 + y dy

dx+ x = 0

(c) d2ydx2 − dy

dx+ x3 = 0

The equation in part (a) is a nonlinear, third-order ordinary differential equation

due to the y2 term. The equation in part (b) is nonlinear due to the y dydx

term. The

equation in part (c), however is linear, despite the x3 term. In this paper, we will

use a model that is a system of ordinary differential equations.

Definition 2.13. A system of ordinary differential equations is a system of

the form:

y′1 = f1(t, y1, y2, ..., yn)

y′2 = f2(t, y1, y2, ..., yn)

...

y′n = fn(t, y1, y2, ..., yn)

(2)

Remark 1. In this case, the order of the system is defined as the number of

equations in the system.

Generally, in real-world applications, there are lots of unknowns within a given

model. The complexity that this brings makes it often difficult or impossible to

solve the system of equations. However, it may still be possible to understand the

behavior of a particular solution within that model. In this paper, we will discuss a

7

three-dimensional, non-linear differential model. So consider the following non-linear

model:

x′ = f(x, y, z)

y′ = g(x, y, z)

z′ = h(x, y, z)

(3)

Definition 2.14. A point (x0, y0, z0) such that

f(x0, y0, z0) = 0

g(x0, y0, z0) = 0

h(x0, y0, z0) = 0

(4)

is said to be a critical point of (6).

Remark 2. Sometimes, in real-world examples, such as population models, it only

makes sense in the 3-dimensional case for the co-existence critical point to occur in

the first octant. In other words, the population sizes must all be positive since it

does not make sense to have a negative population size.

The analysis of this paper will attempt to promote understanding of a specific

population model through understanding of the behavior of the critical points in the

model.

Definition 2.15. A critical point (x0, y0, z0) of (6) is stable if ∀ε > 0, ∃δ > 0 such

that if any solution (x(t), y(t)) of (6) satisfies |x(t0)−x0|+|y(t0)−y0|+|z(t0)−z0| < δ,

then it also satisfies |x(t)− x0|+ |y(t)− y0|+ |z(t)− z0| < ε∀t ≥ t0.

Definition 2.16. Critical points that are not stable are said to be unstable.

Definition 2.17. A critical point (x0, y0, z0) is asymptotically stable if it is stable

and if ∃η > 0 such that if any solution of (6) that satisfies |x(t0) − x0| + |y(t0) −

y0|+ |z(t0)− z0| < η also satisfies limt→∞|x(t)− x0|+ |y(t)− y0|+ |z(t)− z0| = 0.

8

Definition 2.18. Given a set of equations:

y1 = f1(x1, x2, x3)

y2 = f2(x1, x2, x3)

y3 = f3(x1, x2, x3),

(5)

The Jacobian Matrix also called “The Jacobian,” is defined by:

dy1dx1

dy1dx2

dy1dx3

dy2dx1

dy2dx2

dy2dx3

dy3dx1

dy3dx2

dy3dx3

(6)

Example 2.19. Consider the following system:

f = 2x+ 3y − 5z

g = 3x+ 4y + 2z

h = 7x+ 2y + z,

(7)

Let the critical point be at the origin (trivial solution). Then the Jacobian matrix

is: 2 3 −5

3 4 2

7 2 1

(8)

Lemma 2.20. A solution of (3) is asymptotically stable at a critical point if and

only if all of the eigenvalues of the Jacobian of (5) evaluated at the critical point

have negative real parts.

Lemma 2.21. If one eigenvalue of the Jacobian of (3) evaluated at the critical point

has a positive real part, then the solution of (5) is unstable at the critical point.

9

Example 2.22. Considering the following Jacobian matrices evaluated at some

critical point.

(a)

−1 1 0

0 −4 2

0 −1 −1

The eigenvalues of this matrix are −1, −2 and −3. So, by Lemma 2.20, the

system is locally asymptotically stable at the critical point.

(b)

1 1 0

0 −3 1

0 0 −1

The eigenvalues of this matrix are −3, −1 and 1. Since one of the eigenval-

ues is a positive, real number, by Lemma 2.21, the system is unstable at the

critical point.

Sometimes, it may be difficult to calculate the eigenvalues of the Jacobian due

to unknown constants in the system. However, it still may be possible using the

Routh-Hurwitz criterion for a 3-dimensional system.

Theorem 2.23 (Routh-Hurwitz Criterion). A solution is asymptotically stable at a

critical point if the characteristic polynomial, of the Jacobian evaluated at the critical

point, given by:

f(x) = λ3 + A1λ2 + A2λ+ A3 (9)

satisfies the following conditions:

• A1 > 0

10

• A3 > 0

• A1A2 − A3 > 0

Corollary 2.1. It is also useful, when applying the Routh-Hurwitz criterion, to

consider the extended Descartes’ rule of signs. The characteristic polynomial given

in (12) is equivalent to the following:

f(x) = λ3 − trAλ2 +1

2[(trA)2 − tr(A2)]λ− detA (10)

where A is the Jacobian of (6).

11

3 A NON-STAGE STRUCTURED MODEL

First, to emphasize the importance of stage-structure within our model, let us

first look at a variation of model (1) that eliminates the stage-structured aspect of

the prey. That is: our new model, which can be thought of as a ”control” model, is

described by the following system of equations:

dN

dt= bN − dN − sNPH(eshN)

dP

dt= P ((esN) ∗H(eshN)−D)

(11)

where the variables, functions and parameters are defined as follows: N is the

population density of the prey, P is the population density of the predator, b is the

per capita birth rate of the prey, which is presented as a constant here since the

model assumes density-independent prey growth, d are parameters giving the per

capita death rates of juvenile and adult prey, D is the per capita death rate of the

predator, s is the per capita capture rate of prey by a searching predator, h is the

predator’s handling time of prey and e represents the relative nutritional value of

an individual prey to the predator. H is a function describing predator satiation

defined as: H(N) = 11+N

One can see the similarities and differences between this model and the Lotka-

Volterra model mentioned in the introduction. It is worth noting that the eigenvalues

of the Jacobian evaluated at the coexistence critical point for the Lotka-Volterra

model are:

• λ1 = i√αγ

• λ2 = −i√αγ

12

Since both eigenvalues are purely imaginary, no conclusion can be drawn regarding

the stability of the system through linear analysis.

We will now move on to analyzing our control model. As in our stage-structured

model, we will consider three possible equibriums: the extinction of both predator

and prey, the extinction of just the predator, and coexistence. Let us begin our

analysis with the trivial equilibrium.

Theorem 3.1. The trivial equilibrium (0, 0) of Model (14) is

• asymptotically stable if b− d < 0

• unstable if b > d

Proof.

Let:

y1 =dN

dt

y2 =dP

dt

(12)

The Jacobian matrix for the (0, 0) equilibrium in Model (14)is:

J(N,P ) =

dy1dN

dy1dP

dy2dN

dy2dP

where

dy1dN

=∂

∂N(bN − dN − sNP

1 + eshN)

= b− d− (1 + eshN)sP − sNP (esh)

(1 + eshN)2

= b− d

dy1dP

=∂

∂P(bN − dN − sNP

1 + eshN)

13

= − sN

1 + eshN

= 0

dy2dN

=∂

∂N(P (

esN

1 + eshN−D))

=(1 + eshN)(esP )− PesN(esh)

(1 + eshN)2

= 0

dy2dP

=∂

∂N(P (

esN

1 + eshN−D))

=esN

1 + eshN−D

= −D

So, the Jacobian matrix of the system (14) for the equilibrium (0, 0), with extinction

of both predator and prey, is:

J(0, 0) =

(b− d) 0

0 −D

The characteristic equation of J(0, 0) is:

|J(0, 0)− λI| =

∣∣∣∣∣∣∣b− d− λ 0

0 −D − λ

∣∣∣∣∣∣∣= (b− d− λ)(−D − λ) = 0

Solving for λ the eigenvalues are:

λ1 = −D

λ2 = b− d

(13)

14

It is clear that lambda1 is always negative and lambda2 will be negative as long

as b− d < 0. This analysis completes the details of the proof. �

We will next consider the critical point (N, 0) where the predator becomes ex-

tinct.

Theorem 3.2. There are infinitely many equilibria of the form (N, 0) of Model (14)

and they exist if

b− d = 0 (14)

Proof. We can solve for N using the following equation derived from Model (14):

bN − dN = 0

Factoring we have:

N(b− d) = 0 ⇒ (b− d) = 0

As we see, in order for this system to have a critical point of the form (N, 0), we

must require that b− d = 0. We also note that there will be infinitely many of these

equilibrium points, each of which can be at most stable. �

The last equilibrium to consider is where both the predator and prey can coexist.

Theorem 3.3. A positive equilibrium of the form (N,P ) will exist if:

• 1−Dh > 0

• b− d > 0

15

Proof. The population densities in the coexistence equilibrium (N,P ) of (14) can

be solved by the following system of algebraic equations that are derived from the

two-dimensional model:

bN − dN − sNP

1 + eshN= 0

esN

1 + eshN−D = 0

(15)

Using the second equation in the system, we can solve for N :

esN

1 + eshN−D = 0

⇒ esN

1 + eshN= D

⇒ esN = D +DeshN

⇒ esN −DeshN = D

⇒ N(es(1−Dh)) = D

⇒ N =D

es(1−Dh)

(16)

In order for N > 0, it is obvious that we require 1 −Dh > 0. Now, we can plug in

16

our equilibrium value for N into the first equation to solve for P:

bN − dN − sNP

1 + eshN= 0

⇒ bD

es(1−Dh)− d

D

es(1−Dh)−

sD

es(1−Dh)P

1 + eshD

es(1−Dh)

= 0

⇒ D(b− d)

es(1−Dh)−

(DP

e(1−Dh))

1 +Dh

1−Dh

= 0

⇒ D(b− d)

es(1−Dh)−

(DP

e(1−Dh))

(1

1−Dh)

= 0

⇒ D(b− d)

es(1−Dh)− DP

e= 0

⇒ D(b− d)

es(1−Dh)=

DP

e

⇒ P =b− d

s(1−Dh)

(17)

Since, we already required 1−Dh > 0 in order for N > 0, we must also require that

b− d > 0 in order for P > 0 �

Theorem 3.4. The coexistence equilibrium (N,P ) = (D

es(1−Dh),

b− d

s(1−Dh)) is

unstable

Proof.

Let:

y1 =dN

dt

y2 =dP

dt

(18)

17

The Jacobian matrix for this equilibrium in model (14)is:

J(N,P ) =

dy1dN

dy1dP

dy2dN

dy2dP

where

dy1dN

=∂

∂N(bN − dN − sNP

1 + eshN)

= b− d− (1 + eshN)sP − sNP (esh)

(1 + eshN)2

= b− d−(1 + esh

D

es(1−Dh))s(b− d)

s(1−Dh)− sD

es(1−Dh)

b− d

s(1−Dh)esh

(1 + eshD

es(1−Dh))2

= b− d−(1 +

Dh

1−Dh)b− d

1−Dh− Dh(b− d)

(1−Dh)2

(1 +Dh

1−Dh)2

= b− d−(

1

1−Dh)2(b− d)(1−Dh)

(1−Dh)2

= (b− d)(1− (1−Dh))

= (b− d)Dh

dy1dP

=∂

∂P(bN − dN − sNP

1 + eshN)

= − sN

1 + eshN

= −s

D

es(1−Dh)

1 + eshD

es(1−Dh)

= −(

D

e(1−Dh))

1 +Dh

1−Dh

18

= −(

D

e(1−Dh))

(1

1−Dh)

= −D

edy2dN

=∂

∂N(P (

esN

1 + eshN−D))

=(1 + eshN)(esP )− PesN(esh)

(1 + eshN)2

=

(1 + eshD

es(1−Dh))(

es(b− d)

s(1−Dh))− b− d

s(1−Dh)

esD

es(1−Dh)esh

(1 + eshD

es(1−Dh))2

=

(1 +Dh

1−Dh)(e(b− d)

1−Dh)− e(b− d)Dh

(1−Dh)2

(1 +Dh

1−Dh)2

=(

1

1−Dh)2e(b− d)(1−Dh)

(1

1−Dh)2

= e(b− d)(1−Dh)

dy2dP

=∂

∂N(P (

esN

1 + eshN−D))

=esN

1 + eshN−D

=

esD

es(1−Dh)

1 + eshD

es(1−Dh)

−D

=(

D

1−Dh)

1 +D

1−Dh

−D

=(

D

1−Dh)

(1

1−Dh)−D

= D −D

19

= 0

So, the Jacobian matrix of the system (14) for the equilibrium (D

es(1−Dh),

b− d

s(1−Dh)),

with coexistence of both predator and prey, is:

J(D

es(1−Dh),

b− d

s(1−Dh)) =

(b− d)Dh −D

e

e(b− d)(1−Dh) 0

The characteristic equation of J(D

es(1−Dh),

b− d

s(1−Dh)) is:

|J( D

es(1−Dh),

b− d

s(1−Dh))− λI| =

∣∣∣∣∣∣∣(b− d)Dh− λ −D

e

e(b− d)(1−Dh) −λ

∣∣∣∣∣∣∣= λ2 − (b− d)Dhλ+

D

ee(b− d)(1−Dh)

= λ2 − (b− d)Dhλ+D(b− d)(1−Dh)

Solving for λ the eigenvalues are:

λ1 =1

2(Dh(b− d) +

√(Dh(b− d))2 − 4D(b− d)(1−Dh))

λ2 =1

2(Dh(b− d)−

√(Dh(b− d))2 − 4D(b− d)(1−Dh))

(19)

Since b−d > 0 is required for our equilibrium values, the real part of λ1 is always

positive. Also, since 1−Dh > 0, the argument inside the square root of λ2 will be

less than (Dh(b−d))2. So λ2 is always positive as well. Therefore, this critical point

is always unstable. �

This emphasizes the importance of a stage-structured model. In the general

Lotka-Volterra model, linear analysis results in no conclusions regarding asymptotic

20

stability of the coexistence critical point. In our control model, linear analysis re-

veals instability of the coexistence critical point everywhere. In our stage-structured

model, however, we will see that there do exist conditions that allow for asymptotic

stability of the coexistence equilibrium.

21

4 THE ULTIMATE BOUNDS FOR THE POPULATIONS

In this section we obtain some preliminary results on upper-bound functions for

the predator and prey populations. These bounds will provide crucial information

on extinction and exponential convergence of the species. In general, the population

dynamics in the stage-structured predator-prey model (1) depends on many ecolog-

ical parameters. However, in this section we see that the relation between a couple

parameters can determine the ultimate ecological outcome of the system.

4.1 Extinction of the predator by death rate and prey handling time

Our first result is about the ultimate bound of the predator population.

Theorem 4.1. If Dh1 > 1 and Dh2 > 1, then the predator population in model (1)

will go to extinction exponentially.

• 0 ≤ P (t) ≤ P (0)e−(

Dh−1

h

)t, with h = min{h1, h2}.

Proof. From the third equation in (1),

dP

dt= P

(e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D

)≤ P

(e1s1N1 + e2s2N2

1 + h(e1s1N1 + e2s2N2)−D

)≤ P

(e1s1N1 + e2s2N2

h(e1s1N1 + e2s2N2)−D

)= P

(1

h−D

)= P

(1−Dh

h

)(20)

where h = min{h1, h2}. This implies that

0 ≤ P (t) ≤ P (0)e

(1−Dh

h

)t.

22

�

4.2 Extinction of the total population by birth and death rate of the adult prey

Theorem 4.2. If d2 > b2, then all three populations in model (1) will go to extinction

exponentially.

• 0 ≤ N1(t) +N2(t) + P (t) ≤ [N1(0) +N2(0) + P (0)]e−δt,

with δ = min{d2 − b2, d1, D} > 0.

In this case, the trivial equilibrium (0, 0, 0) is globally asymptotically stable.

Proof. Let F (t) = N1(t)+N2(t)+P (t) be the total population of the three species

in model (1). Based on the fact that the relative nutritional values for the preys

0 < e1, e2 ≤ 1, the summation of the three equations in (1) gives:

dF

dt= (b2 − d2)N2 − d1N1 − P

((1− e1)s1N1 + (1− e2)s2N2

1 + e1s1h1N1 + e2s2h2N2

+D

)≤ (b2 − d2)N2 − d1N1 −DP

≤ −δF

(21)

where δ = min{d2 − b2, d1, D}. This implies that

0 ≤ F (t) ≤ F (0)e−δt.

�

23

5 EQUILIBRIUM POINTS WITH EXTINCTION OF PREDATOR

In this section, we give stability results on the two equilibrium points with ab-

sence of the predator population.

5.1 Equilibrium for extinction of predator and prey

We will first consider the trivial equilibrium, that is, the equilibrium where both

the predator and prey become extinct.

Theorem 5.1. The trivial equilibrium (0, 0, 0) is

• asymptotically stable if b2g < d2(d1 + g),

• unstable if b2g > d2(d1 + g).

Proof.

Let:

y1 =dN1

dt

y2 =dN2

dt

y3 =dP

dt

(22)

The Jacobian matrix for this equilibrium in model (1)is:

J(N1, N2, P ) =

dy1dN1

dy1dN2

dy1dP

dy2dN1

dy2dN2

dy2dP

dy3dN1

dy3dN2

dy3dP

24

where

dy1dN1

=∂

∂N1

(b2N2 − d1N1 − gN1 −s1N1

1 + e1s1h1N1 + e2s2h2N2

)

= −d1 − g − s1P (1 + e2s2h2N2)

(1 + e1s1h1N1 + e2s2h2N2)2

= −d1 − g

dy1dN2

=∂

∂N2

(b2N2 − d1N1 − gN1 −s1N1

1 + e1s1h1N1 + e2s2h2N2

)

= b2 +e2s1s2h2N1P

(1 + e1s1h1N1 + e2s2h2N2)2

= b2

dy1dP

=∂

∂P(b2N2 − d1N1 − gN1 −

s1N1

1 + e1s1h1N1 + e2s2h2N2

)

= − s1N1

1 + e1s1h1N1 + e2s2h2N2

= 0

dy2dN1

=∂

∂N1

(gN1 − d2N2 −s2N2

1 + e1s1h1N1 + e2s2h2N2

)

= g +e1s1s2h1N2P

(1 + e1s1h1N1 + e2s2h2N2)2

= g

dy2dN2

=∂

∂N2

(gN1 − d2N2 −s2N2

1 + e1s1h1N1 + e2s2h2N2

)

= −d2 −s2P (1 + e1s1h1N1)

(1 + e1s1h1N1 + e2s2h2N2)2

= −d2

dy2dP

=∂

∂P(gN1 − d2N2 −

s2N2

1 + e1s1h1N1 + e2s2h2N2

)

= − s2N2

1 + e1s1h1N1 + e2s2h2N2

= 0

dy3dN1

=∂

∂N1

(P (e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D))

=e1s1P [1 + e2s2(h2 − h1)N2]

(1 + e1s1h1N1 + e2s2h2N2)2

25

= 0

dy3dN2

=∂

∂N2

(P (e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D))

=e2s2P [1 + e1s1(h1 − h2)N2]

(1 + e1s1h1N1 + e2s2h2N2)2

= 0

dy3dP

=∂

∂P(P (

e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D))

=e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D

= −D

So, the Jacobian matrix of the system (1) for the equilibrium (0, 0, 0), with

extinction of both predator and prey, is:

J(0, 0, 0) =

−d1 − g b2 0

g −d2 0

0 0 −D

The characteristic equation of J(0, 0, 0) is:

|J(0, 0, 0)− λI| =

∣∣∣∣∣∣∣∣∣∣−d1 − g − λ b2 0

g −d2 − λ 0

0 0 −D − λ

∣∣∣∣∣∣∣∣∣∣= (−D − λ)

∣∣∣∣∣∣∣−d1 − g − λ b2

g −d2 − λ

∣∣∣∣∣∣∣= (−D − λ)((−d1 − g − λ)(−d2 − λ)− b2g) = 0

26

Solving for λ we have:

(−D − λ)((−d1 − g − λ)(−d2 − λ)− b2g) = 0

⇒ (−D − λ)(d1d2 + d1λ+ gd2 + gλ+ d2λ+ λ2 − b2g) = 0

⇒ (−D − λ)(λ2 + (d1 + g + d2)λ+ d1d2 + gd2 − b2g) = 0

⇒ (−D − λ) = 0 or (λ2 + (d1 + g + d2)λ+ d1d2 + gd2 − b2g) = 0

⇒ λ = −D or λ =−d1 − g − d2 ±

√(d1 + g + d2)2 − 4(d1d2 + gd2 − b2g)

2

So, the eigenvalues for this matrix are:

• λ1 = −D

• λ2 =12(−

√(d1 + d2 + g)2 − 4(d1d2 + d2g − b2g)− d1 − d2 − g)

• λ3 =12(√(d1 + d2 + g)2 − 4(d1d2 + d2g − b2g)− d1 − d2 − g)

In order for the trivial equilibrium to be asymptotically stable, all three of the

eigenvalues must have a negative real part. Since D > 0 is per capita predator death

rate, λ1 > 0. We next turn to conditions ensuring λ2 and λ3 both have negative real

parts.

If 0 < 4(d1d2 + d2g − b2g) < (d1 + d2 + g)2, then the argument inside the radical

for both λ2 and λ3 are negative. This way both λ2 and λ3 will be complex with

negative real part (−d1− d2− g)/2. If 0 < (d1+ d2+ g)2 ≤ 4(d1d2+ d2g− b2g), then

both λ2 and λ3 are negative real numbers. One of these conditions can be ensured

when b2g < d2(d1 + g). On the other hand, if b2g > d2(d1 + g) then λ3 is a positive

real number. �

27

5.2 Equilibrium for extinction of predator only

We will next consider the equilibrium point with extinction of the predator

(N1, N2, 0).

Theorem 5.2. There are infinitely many equilibrium of the form (N1, N2, 0) of

Model (1) and they exist iff

b2g ̸= d2(d1 + g) (23)

Proof. The positive population sizes N1 and N2 must satisfy:

• b2N2 − d1N1 − gN1 = 0 and gN1 − d2N2 = 0.

This is a system of dependent equations so there will be an infinite number of solu-

tions. Solving for N1 using the first equation we get: N1 =b2N2

d1+g. Solving for N1 in

the second equation we get N1 =d2N2

g. So, we must have that b2g = d2(d1 + g). The

prey-only equilibrium ( b2N2

d1+g, N2, 0) does not exist if b2g ̸= d2(d1 + g). �

Theorem 5.3. When b2g = d2(d1 + g), the system (1) has infinite many prey-only

equilibria (N1,gN1

d2, 0). Each equilibrium in this set is:

• stable if e1s1b2N2(1−Dh1) + e2s2N2(d1 + g)(1−Dh2) ≤ D(d1 + g),

• unstable if e1s1b2N2(1−Dh1) + e2s2N2(d1 + g)(1−Dh2) > D(d1 + g).

Proof.

28

Let:

y1 =dN1

dt

y2 =dN2

dt

y3 =dP

dt

(24)

The Jacobian matrix for this equilibrium in model (1)is:

J(N1, N2, P ) =

dy1dN1

dy1dN2

dy1dP

dy2dN1

dy2dN2

dy2dP

dy3dN1

dy3dN2

dy3dP

where

dy1dN1

=∂

∂N1

(b2N2 − d1N1 − gN1 −s1N1

1 + e1s1h1N1 + e2s2h2N2

)

= −d1 − g − s1P (1 + e2s2h2N2)

(1 + e1s1h1N1 + e2s2h2N2)2

= −d1 − g

dy1dN2

=∂

∂N2

(b2N2 − d1N1 − gN1 −s1N1

1 + e1s1h1N1 + e2s2h2N2

)

= b2 +e2s1s2h2N1P

(1 + e1s1h1N1 + e2s2h2N2)2

= b2

dy1dP

=∂

∂P(b2N2 − d1N1 − gN1 −

s1N1

1 + e1s1h1N1 + e2s2h2N2

)

= − s1N1

1 + e1s1h1N1 + e2s2h2N2

= −s1

b2N2

d1 + g

1 + e1s1h1b2N2

d1 + g+ e2s2h2N2

= −

s1b2N2

d1 + gd1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g

d1 + g

29

= − s1b2N2

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g

dy2dN1

=∂

∂N1

(gN1 − d2N2 −s2N2

1 + e1s1h1N1 + e2s2h2N2

)

= g +e1s1s2h1N2P

(1 + e1s1h1N1 + e2s2h2N2)2

= g

dy2dN2

=∂

∂N2

(gN1 − d2N2 −s2N2

1 + e1s1h1N1 + e2s2h2N2

)

= −d2 −s2P (1 + e1s1h1N1)

(1 + e1s1h1N1 + e2s2h2N2)2

= −d2

dy2dP

=∂

∂P(gN1 − d2N2 −

s2N2

1 + e1s1h1N1 + e2s2h2N2

)

= − s2N2

1 + e1s1h1N1 + e2s2h2N2

= − s2N2

1 + e1s1h1b2N2

d1+g+ e2s2h2N2

= − s2N2

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g

d1 + g

= − s2N2(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g

dy3dN1

=∂

∂N1

(P (e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D))

=e1s1P [1 + e2s2(h2 − h1)N2]

(1 + e1s1h1N1 + e2s2h2N2)2

= 0

dy3dN2

=∂

∂N2

(P (e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D))

=e2s2P [1 + e1s1(h1 − h2)N2]

(1 + e1s1h1N1 + e2s2h2N2)2

= 0

dy3dP

=∂

∂P(P (

e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D))

=e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D

30

=e1s1

b2N2

d1+g+ e2s2N2

1 + e1s1h1b2N2

d1+g+ e2s2h2N2

−D

=

e1s1b2N2 + e2s2N2d1 + e2s2N2g

d1 + gd1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g

d1 + g

−D

=e1s1b2N2 + e2s2N2d1 + e2s2N2g

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g−D

=e1s1b2N2(1−Dh1) + e2s2N2(d1 + g)(1−Dh2)−D(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g

So, assuming b2g = d2(d1 + g), the Jacobian Matrix for (N1, N2, 0) is:

−d1 − g b2 − s1b2N2

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g

g −d2 − s2N2(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g

0 0e1s1b2N2(1−Dh1) + e2s2N2(d1 + g)(1−Dh2)−D(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g

The characteristic equation of J( b2N2

d1+g, N2, 0) is:

|J(b2N2

d1 + g,N2, 0)− λI| =∣∣∣∣∣∣∣∣∣∣∣

−d1 − g − λ b2 −s1b2N2

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g

g −d2 − λ −s2N2(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g

0 0e1s1b2N2(1−Dh1) + e2s2N2(d1 + g)(1−Dh2)−D(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g− λ

∣∣∣∣∣∣∣∣∣∣∣= (

e1s1b2N2(1−Dh1) + e2s2N2(d1 + g)(1−Dh2)−D(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g− λ)

∣∣∣∣∣∣−d1 − g − λ b2

g −d2 − λ

∣∣∣∣∣∣= (

e1s1b2N2(1−Dh1) + e2s2N2(d1 + g)(1−Dh2)−D(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g− λ)((−d1 − g − λ)(−d2 − λ)− b2g) = 0

31

Solving for λ we have:

(e1s1b2N2(1−Dh1) + e2s2N2(d1 + g)(1−Dh2)−D(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g− λ)((−d1 − g − λ)(−d2 − λ)− b2g) = 0

⇒ (e1s1b2N2(1−Dh1) + e2s2N2(d1 + g)(1−Dh2)−D(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g− λ)(d1d2 + d1λ+ gd2 + gλ+ d2λ+ λ2 − b2g) = 0

⇒ (e1s1b2N2(1−Dh1) + e2s2N2(d1 + g)(1−Dh2)−D(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g− λ)(λ2 + (d1 + g + d2)λ+ d1d2 + gd2 − b2g) = 0

⇒ (e1s1b2N2(1−Dh1) + e2s2N2(d1 + g)(1−Dh2)−D(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g− λ) = 0

or (λ2 + (d1 + g + d2)λ+ d1d2 + gd2 − b2g) = 0

⇒ λ =e1s1b2N2(1−Dh1) + e2s2N2(d1 + g)(1−Dh2)−D(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g

or λ =−d1 − g − d2 ±

√(d1 + g + d2)2 − 4(d1d2 + gd2 − b2g)

2

So, the eigenvalues of this matrix are:

• λ1 =1

2(−

√(d1 + d2 + g)2 − 4(−b2g + d1d2 + d2g)− d1 − d2 − g)

• λ2 =1

2(√(d1 + d2 + g)2 − 4(−b2g + d1d2 + d2g)− d1 − d2 − g)

• λ3 =e1s1b2N2(1−Dh1) + e2s2N2(d1 + g)(1−Dh2)−D(d1 + g)

d1 + g + e1s1h1b2N2 + e2s2h2N2d1 + e2s2h2N2g

Under the assumption of b2g − d1d2 − d2g = 0, there are an infinite number of

equilibria and asymptotic stability cannot occur. We can also obtain that λ1 =

−(d1 + d2 + g) < 0 and λ2 = 0. Since lamda2 = 0, linear analysis cannot tell us

anything regarding the stability of the system. However, since the equilibrium points

form a line, we can conclude that they each must be stable.

So, the conclusion of the theorem comes directly from λ3 �

We now illustrate the set of stable equilibria (N1, N2, P ) through numerical sim-

ulations of model (1), with parameters: {b2 = 0.8, g = 0.1, d1 = 0.3, d2 = 0.2, e1 =

0.5, e2 = 1.0, s1 = 0.6, s2 = 0.3, h1 = 0.1, h2 = 0.2, D = 0.1}. These ecological pa-

rameters satisfy b2g = d2(d1+g), so the model has an infinite set of prey-only equilib-

ria in the form of (N1, N1/2, 0). With initial populations N1(0) = 0.2, N2(0) = 0.3,

32

P (0) = 0.1, the numerical simulation of model (1) demonstrated in Figure 1 shows

the convergence to an equilibrium (0.0536352788, 0.0268176394, 0).

Figure 1: Stability of Prey-only Equilibrium

Changing the initial populations as N1(0) = 0.25, N2(0) = 0.2, P (0) = 0.3, the

numerical simulation of model (1) demonstrated in Figure 2 shows the convergence

to another equilibrium (0.0278454640, 0.0139227320, 0).

Figure 2: Stability of Prey-only Equilibrium

33

6 COEXISTENCE WHILE ONLY ONE STAGE OF THE PREY IS

VULNERABLE

We now consider the conditions for both the prey and predator to coexist in

model (1). First, we will examine a couple special cases where the prey is vulnerable

to the predator at only one stage. After that, we then proceed to the general case.

6.1 Juveniles are vulnerable

Often, in biological systems, the predator is similar in size to the prey, and so

attacking adult prey can actually be harmful to the predator. An example of this is

the perch which will only consume juvenile, or immature prey [5]. Therefore, in these

systems, the predator will only attack juvenile prey. The first case considered here

assumes that only the juvenile prey are vulnerable to the predator and is described

by the following equations: [1]

dN1

dt= b2N2 − d1N1 − gN1 − sN1P ∗H(eshN1)

dN2

dt= gN1 − d2N2

dP

dt= P (esN1 ∗H(eshN1)−D)

(25)

Here theH function only takes eshN1 as its parameter and we can drop the subscript

1 from e1, s1, and h1.

Theorem 6.1. The conditions for the existence of a positive equilibrium of the form

N1, N2, P of Model (30) are:

0 < Dh < 1 and gb2 − d2g − d1d2 > 0. (26)

Proof. The population densities in the coexistence equilibrium (N1, N2, P ) of (30)

can be solved by the following system of algebraic equations that are derived from

34

model (30):

b2N2 − d1N1 − gN1 −sN1P

1 + eshN1

= 0

gN1 − d2N2 = 0

esN1

1 + eshN1

−D = 0

(27)

Using the third equation in the system, we can solve for N1:

esN1

1 + eshN1

−D = 0

⇒ esN1

1 + eshN1

= D

⇒ esN1 = D +DeshN1

⇒ esN1 −DeshN1 = D

⇒ N1(es(1−Dh)) = D

⇒ N1 =D

es(1−Dh)

(28)

Now, using the second equation in the system, and our equilibrium value for N1, we

can solve for N2:

gN1 − d2N2 = 0

⇒ gD

es(1−Dh)− d2N2 = 0

⇒ gD

es(1−Dh)= d2N2

⇒ N2 =gD

d2es(1−Dh)

(29)

Finally, using the first equation in the system and our equilibrium values for N1 and

35

N2, we can solve for P :

b2N2 − d1N1 − gN1 −sN1P

1 + eshN1

= 0

⇒ b2gD

d2es(1−Dh)− D(d1 + g)

es(1−Dh)−

sD

es(1−Dh)P

1 + eshD

es(1−Dh)

= 0

⇒ b2gD − d2D(d1 + g)

d2es(1−Dh)=

sD

es(1−Dh)P

1 + eshD

es(1−Dh)

⇒ P =

[1 + esh(D

es(1−Dh))][b2gD − d2D(d1 + g)]

d2es2(1−Dh)(D

es(1−Dh))

⇒ P =(1 +

Dh

1−Dh)[D(b2g − d2(d1 + g))]

d2Ds

⇒ P =(g(b2 − d2)− d1d2)

d2s(1−Dh)

(30)

In summary, the population densities in the coexistence equilibrium (N1, N2, P ) of

(30) are:

N1 =D

es(1−Dh)

N2 =Dg

d2es(1−Dh)

P =g(b2 − d2)− d1d2d2s(1−Dh)

(31)

In order for N1 and N2 to be positive, we must require that 1 − Dh > 0.

Due to this requirement, in order for P to be positive, we must also require that

gb2 − d2g − d1d2 > 0. �

Theorem 6.2. For the model (30) where only the juvenile prey is vulnerable, there

is a unique coexistence equilibrium (N1, N2, P ) given by (36) if the conditions in (31)

36

hold. This equilibrium is asymptotically stable if:

(1−Dh− d2h)(gb2 − gb2Dh+ gd2Dh+ d1d2Dh) > d32h, (32)

and unstable if

(1−Dh− d2h)(gb2 − gb2Dh+ gd2Dh+ d1d2Dh) < d32h.

Proof.

Let:

y1 =dN1

dt

y2 =dN2

dt

y3 =dP

dt

(33)

The Jacobian matrix for this equilibrium in model (30)is:

J(N1, N2, P ) =

dy1dN1

dy1dN2

dy1dP

dy2dN1

dy2dN2

dy2dP

dy3dN1

dy3dN2

dy3dP

where:

dy1dN1

=∂

∂N1

(b2N2 − d1N1 − gN1 −sN1P

1 + eshN1

)

= −d1 − g − [(1 + eshN1)sP − sN1P (esh)]

(1 + eshN1)2

= −d1 − g − sP + es2hN1P − es2hN1P

(1 + eshN1)2

= −d1 − g − sP

(1 + eshN1)2

37

= −d1 − g −

s[g(b2 − d2)− d1d2]

d2s(1−Dh)

(1 + eshD

es(1−Dh))2

= −d1 − g −

g(b2 − d2)− d1d2d2(1−Dh)

(1 +Dh

1−Dh)2

= −d1 − g −

g(b2 − d2)− d1d2d2(1−Dh)

1

(1−Dh)2

= −d1 − g − [g(b2 − d2)− d1d2](1−Dh)

d2

=−d1d2 − gd2 − gb2 + gd2 + d1d2 + gb2Dh− gd2Dh− d1d2Dh

d2

=−b2g + b2gDh− d1d2Dh− d2gDh

d2dy1dN2

=∂

∂N2

(b2N2 − d1N1 − gN1 −sN1P

1 + eshN1

)

= b2

dy1dP

=∂

∂P(b2N2 − d1N1 − gN1 −

sN1P

1 + eshN1

)

=−sN1

1 + eshN1

= −

sD

es(1−Dh)

1 + esh(D

es(1−Dh))

= −(

D

e(1−Dh))

1 +Dh

1−Dh

= −(

D

e(1−Dh))

(1

1−Dh)

= −D

edy2dN1

=∂

∂N1

(gN1 − d2N2)

38

= g

dy2dN2

=∂

∂N1

(gN1 − d2N2)

= −d2

dy2dP

=∂

∂N1

(gN1 − d2N2)

= 0

dy3dN1

=∂

∂N1

(P (esN1

1 + eshN1

−D)

=(1 + eshN1)esP − esN1P (esh)

(1 + eshN1)2

=

(1 +eshD

es(1−Dh))es(g(b2 − d2)− d1d2)

d2s(1−Dh)− esD

es(1−Dh)∗ esh(g(b2 − d2)− d1d2)

d2s(1−Dh)

(1 + eshD

es(1−Dh))2

=

(1 +Dh

1−Dh)eg(b2 − d2)− d1d2

d2(1−Dh)− D

1−Dh∗ g(b2 − d2)− d1d2

d2(1−Dh)eh

(1 +Dh

1−Dh)2

=(

1

1−Dh)2[e

g(b2 − d2)− d1d2d2

− eDhg(b2 − d2)− d1d2

d2]

(1

1−Dh)2

=e(g(b2 − d2)− d1d2)(1−Dh)

d2

=e(1−Dh)(b2g − d1d2 − d2g)

d2dy3dN2

=∂

∂N2

(P (esN1

1 + eshN1

−D)

= 0

dy3dP

=∂

∂P(P (

esN1

1 + eshN1

−D)

=esN1

1 + eshN1

−D

=

esD

es(1−Dh)

1 + eshD

es(1−Dh)

−D

39

=(

D

1−Dh)

1 +Dh

1−Dh

−D

=(

D

1−Dh)

(1

1−Dh)−D

= D −D

= 0

In summary, the Jacobian matrix for this equilibrium in model (30) is:

J(N1, N2, P ) =

−b2g + b2gDh− d1d2Dh− d2gDh

d2b2 −D

e

g −d2 0

e(1−Dh)(b2g − d1d2 − d2g)

d20 0

The determinant of (J(N1, N2, P )− λI) is: λ3 + A1λ

2 + A2λ+ A3 where:

• A1 =b2g(1−Dh)

d2+ d1Dh+ gDh+ d2

• A2 =D(b2g − d1d2 − d2g)(1−Dh)

d2−Dh(b2g − d1d2 − d2g)

• A3 = D(b2g − d1d2 − d2g)(1−Dh)

Applying the Routh-Hurwitz criteria for stability, the coexistence equilibrium is

asymptotically stable if : A1 > 0, A3 > 0, and A1A2 > A3. We first observe that the

positivity condition (31) for the equilibrium (N1, N2, P ) given in (36) ensures that

A1 > 0 and A3 > 0.

A1A2 − A3 =

D(gb2 − gd2 − d1d2)

d22

[(1−Dh− d2h)(gb2 − gb2Dh+ gd2Dh+ d1d2Dh)− d32h

]40

This analysis concludes the details of the proof. �

In order to show that the coexistence equilibrium will be asymptotically stable

while d2 (the death rate of the adult prey) is sufficiently small, we also observe that

A1 =b2g(1−Dh)

d2+ d1Dh+ gDh+ d2 > d1Dh+ gDh+ d2

⇒ A1A2 − A3 > (d1Dh+ gDh+ d2)A2 − A3

= Dh(b2g − d1d2 − d2g)

[D(d1 + g)(1−Dh)

d2− (d1Dh+ gDh+ d2)

].

Hence we can conclude that A1A2 > A3 if D(d1+g)(1−Dh) > d2(d1Dh+gDh+d2),

which is equivalent to the condition given in the following corollary:

Corollary 6.1. If the conditions in (31) hold, then the unique coexistence equilib-

rium (N1, N2, P ) given by (36) in model (30) is asymptotically stable whenever:

d2 <

√D(d1 + g)(1−Dh) +

[Dh(d1 + g)

2

]2− Dh(d1 + g)

2. (34)

Proof. We know that if the conditions (31) hold, then the unique coexistence equi-

librium (N1, N2, P ) given by (36) in model (30) is asymptotically stable whenever

D(d1 + g)(1−Dh) > d2(d1Dh+ gDh+ d2). So, let:

41

D(d1 + g)(1−Dh) > d2(d1Dh+ gDh+ d2)

⇒ D(d1 + g)(1−Dh) > d22 +Dh(d1 + g)d2

⇒ D(d1 + g)(1−Dh) +

[Dh(d1 + g)

2

]2> d22 +Dh(d1 + g)d2 +

[Dh(d1 + g)

2

]2⇒ D(d1 + g)(1−Dh) +

[Dh(d1 + g)

2

]2> (d2 +

Dh(d1 + g)

2)2

⇒

√D(d1 + g)(1−Dh) +

[Dh(d1 + g)

2

]2> d2 +

Dh(d1 + g)

2

⇒ d2 <

√D(d1 + g)(1−Dh) +

[Dh(d1 + g)

2

]2− Dh(d1 + g)

2.�

(35)

We can also find a necessary condition for asymptotic stability for this this special

case. We first must present a simple algebraic lemma.

Lemma 6.3.√A+B2 −B > 0 ⇒ A

2B>

√A+B2 −B

Proof. √A+B2 −B > 0

⇒ (√A+B2 −B)2 > 0

⇒ A+ 2B2 − 2B√A+B2 > 0

⇒ A+ 2B2 > 2B√A+B2

⇒ A

2B+B >

√A+B2

⇒ A

2B>

√A+B2 −B.�

(36)

Corollary 6.2. h(D + d2) < 1 is a necessary condition for asymptotic stability of

the coexistence equilibrium (N1, N2, P ) given by (36) in model (30)

42

Proof. Applying the results of Corollary 6.1 and Lemma 6.3, we have:

d2 <D(d1 + g)(1−Dh)

2(Dh(d1 + g)

2)

⇒ d2 <1−Dh

h

⇒ d2h < 1−Dh

⇒ h(D + d2) < 1.�

(37)

We now give an example of coexistence in model (30) with numerical simulations.

With the following set of ecological parameters: {b2 = 0.8, g = 0.2, d1 = 0.3, d2 =

0.2, e = 0.5, s = 0.8, h = 0.1, D = 0.1}, the condition for asymptotic stability of

the coexistence equilibrium is ensured by Theorem 6.2. The coexistence equilibrium

given by (36) is

(N1, N2, P ) = (0.3367005567, 0.3367003367, 0.5050505050).

With initial populations N1(0) = 0.2, N2(0) = 0.3, P (0) = 0.1, the numerical

simulation of model (30) demonstrated in Figure 3 shows that all three populations

converge to the equilibrium values for coexistence.

6.2 Adults are vulnerable

Another occurrence in biological systems is when the predator only consumes

adult prey. One reason for this may be that the predator is much larger than its

prey. Therefore, it is not efficient for it to consume young prey. Other reasons

also cause for this type of predation. For example, certain species of cicadas live

underground during their juvenile or immature stage for up to 17 years, and will

come out of ground as adults and live for a couple weeks where they ar eaten by

predators. Since the cicadas are underground as juveniles, they are not vulnerable

43

Figure 3: Asymptotic Stability, Coexistence While Only Juvenile Vulnerable

until their adult stage. Assuming that the predator will only consume adult prey,

the second special case considered here is described by the following equations:

dN1

dt= b2N2 − d1N1 − gN1

dN2

dt= gN1 − d2N2 − sN2P ∗H(eshN2)

dP

dt= P (esN2 ∗H(eshN2)−D)

(38)

Here theH function only takes eshN2 as its parameter and we can drop the subscript

2 from e2, s2, and h2.

Theorem 6.4. The conditions for the existence of a positive equilibrium of the form

N1, N2, P of Model (43) are:

0 < Dh < 1 and gb2 − d2g − d1d2 > 0. (39)

Proof.

The population densities in the coexistence equilibrium (N1, N2, P ) of (43) can

44

be solved by the following system of algebraic equations that are derived from model

(43):

b2N2 − d1N1 − gN1 = 0

gN1 − d2N2 −sN2P

1 + eshN2

= 0

esN2

1 + eshN2

−D = 0

(40)

Using the third equation in the system, we can solve for N2:

esN2

1 + eshN2

−D = 0

⇒ esN2

1 + eshN2

= D

⇒ esN2 = D +DeshN2

⇒ esN2 −DeshN2 = D

⇒ N2(es− eshD) = 0

⇒ N2 =D

es(1−Dh)

(41)

Now, using our equilibrium value for N2 and the first equation in the system, we

can solve for N1:

b2N2 − d1N1 − gN1 = 0

⇒ b2D

es(1−Dh)− d1N1 − gN1 = 0

⇒ b2D

es(1−Dh)− (d1 + g)N1 = 0

⇒ −(d1 + g)N1 = − b2D

es(1−Dh)

⇒ N1 =b2D

es(1−Dh)(d1 + g)

(42)

Finally, using our equilibrium values for N1 and N2 and the second equation in the

45

system, we can solve for P :

gN1 − d2N2 −sN2P

1 + eshN2

= 0

⇒ gb2D

es(1−Dh)(d1 + g)− d2D

es(1−Dh)=

sPD

es(1−Dh)

1 + eshD

es(1−Dh)

⇒ gb2D − (d1 + g)d2D

es(1−Dh)(d1 + g)=

(DP

e(1−Dh))

1 +Dh

1−Dh

⇒e(1−Dh)(1 +

Dh

1−Dh)(gb2D − (d1 + g)d2D)

esD(1−Dh)(d1 + g)= P

⇒(1 +

Dh

1−Dh)(g(b2 − d2)− d1d2)

s(d1 + g)= P

⇒ P =g(b2 − d2)− (d1d2)

s(1−Dh)(d1 + g)

(43)

The population densities in the coexistence equilibrium (N1, N2, P ) of model (43)

are:

N1 =b2D

es(1−Dh)(d1 + g)

N2 =D

es(1−Dh)

P =g(b2 − d2)− (d1d2)

s(1−Dh)(d1 + g)

(44)

It is obvious that the conditions in (31) also ensures the positivity of the equilibrium

(49) for model (43). �

Theorem 6.5. For the model (43) where only the adult prey is vulnerable, there is

a unique coexistence equilibrium (N1, N2, P ) given by (49) if the conditions in (44)

46

hold. This equilibrium is asymptotically stable if:

(1−Dh− d1h− gh)(gb2 − gb2Dh+ gd2Dh+ d1d2Dh) > (d1 + g)3h. (45)

and unstable if

(1−Dh− d1h− gh)(gb2 − gb2Dh+ gd2Dh+ d1d2Dh) < (d1 + g)3h.

Proof.

Let:

y1 =dN1

dt

y2 =dN2

dt

y3 =dP

dt

(46)

The Jacobian matrix for this equilibrium in model (43)is:

J(N1, N2, P ) =

dy1dN1

dy1dN2

dy1dP

dy2dN1

dy2dN2

dy2dP

dy3dN1

dy3dN2

dy3dP

where:

dy1dN1

=∂

∂N1

(b2N2 − d1N1 − gN1)

= −d1 − g

dy1dN2

=∂

∂N2

(b2N2 − d1N1 − gN1)

= b2

dy1dP

=∂

∂P(b2N2 − d1N1 − gN1)

47

= 0

dy2dN1

=∂

∂N2

(gN1 − d2N2 −sN2P

1 + eshN2

)

= g

dy2dN2

=∂

∂N2

(gN1 − d2N2 −sN2P

1 + eshN2

)

= −d2 − [(1 + eshN2)sP − sN2P (esh)

(1 + eshN2)2]

= −d2 − [sP

(1 + eshN2)2]

= −d2 − [sP

(1 + eshDh

es(1−Dh))2]

= −d2 − [

s(g(b2 − d2)− d1d2)

s(1−Dh)(d1 + g)

(1 +eshD

es(1−Dh))2

]

= −d2 − [

g(b2 − d2)− d1d2(1−Dh)(d1 + g)

(1 +Dh

(1−Dh))2

]

= −d2 − [

g(b2 − d2)− d1d2(1−Dh)(d1 + g)

(1

(1−Dh))2

]

= −d2 − [(g(b2 − d2)− d1d2)(1−Dh)

d1 + g]

= −d2 −b2g − gd2 − d1d2 − b2gDh+ gd2Dh+ d1d2Dh

d1 + g

=−d1d2 − gd2 − b2g + gd2 + d1d2 + b2gDh− d2gDh− d1d2Dh

d1 + g

=−b2g + b2gDh− d1d2Dh− d2gDh

d1 + g

dy2dP

=∂

∂P(gN1 − d2N2 −

sN2P

1 + eshN2

)

= − sN2

1 + eshN2

48

= −s(

D

es(1−Dh))

1 + esh(D

es(1−Dh))

= −(

D

e(1−Dh))

1 +Dh

1−Dh

= −(

D

e(1−Dh))

(1

1−Dh)

= −D

edy3dN1

=∂

∂N1

(P (esN2

1 + eshN2

−D))

= 0

dy3dN2

=∂

∂N2

(P (esN2

1 + eshN2

−D))

=(1 + eshN2)esP − esN2P (esh)

(1 + eshN2)2

=

(1 +eshD

es(1−Dh))es(g(b2 − d2)− d1d2)

s(1−Dh)(d1 + g)− esD

es(1−Dh)esh(g(b2−d2)−d1d2)

s(1−Dh)(d1+g)

(1 + esh Des(1−Dh)

)2

=

(1 +Dh

1−Dh)e(g(b2 − d2)− d1d2)

(1−Dh)(d1 + g)− eDh

1−Dh

g(b2 − d2)− d1d2s(1−Dh)(d1 + g)

(1 + Dh1−Dh

)2

=

(1

1−Dh)e(g(b2 − d2)− d1d2)

(1−Dh)(d1 + g)− eDh

1−Dhg(b2−d2)−d1d2s(1−Dh)(d1+g)

( 11−Dh

)2

=( 11−Dh

)2e(g(b2 − d2)− d1d2)(1−Dh)

( 11−Dh

)2

=e(1−Dh)(b2g − d1d2 − d2g)

d1 + g

dy3dP

=∂

∂P(P (

esN2

1 + eshN2

−D))

=esN2

1 + eshN2

−D

49

=es D

es(1−Dh)

1 + esh Des(1−Dh)

−D

=( D1−Dh

)

1 + Dh1−Dh

=( D1−Dh

)

( 11−Dh

)−D

= D −D

= 0

In summary, the Jacobian matrix for this equilibrium in model (43) is:

J(N1, N2, P ) =

−d1 − g b2 0

g−b2g + b2gDh− d1d2Dh− d2gDh

d1 + g−D

e

0e(1−Dh)(b2g − d1d2 − d2g)

d1 + g0

Again, the determinant of (J(N1, N2, P )− λI) is: λ3 + A1λ

2 + A2λ+ A3 where:

• A1 =b2g(1−Dh)

d1 + g+ d1 + g + d2Dh

• A2 =D(b2g − d1d2 − d2g)(1−Dh)

d1 + g−Dh(b2g − d1d2 − d2g)

• A3 = D(b2g − d1d2 − d2g)(1−Dh)

Once again, by applying the Routh-Hurwitz criteria for stability, we see that the

coexistence equilibrium is asymptotically stable if : A1 > 0, A3 > 0, and A1A2 > A3.

The positivity condition (49) for the equilibrium (N1, N2, P ) given in (44) already

ensures that A1 > 0 and A3 > 0.

A1A2 −A3 =

D(gb2 − gd2 − d1d2)

(d1 + g)2[(1−Dh− d1h− gh)(gb2 − gb2Dh+ gd2Dh+ d1d2Dh)− (d1 + g)3h

]50

This analysis concludes the details of the proof. �

For a simplified sufficient condition of asymptotic stability, We can also obtain

that

A1A2 − A3 > (d1 + g + d2Dh)A2 − A3

= Dh(b2g − d1d2 − d2g)

[Dd2(1−Dh)

d1 + g− (d1 + g + d2Dh)

].

From here we conclude that A1A2 > A3 if Dd2(1−Dh) > (d1 + g)(d1 + g + d2Dh),

which can hold if the sum of d1 and g (the death rate of the juvenile and the transition

rate of juvenile prey to adult) is sufficiently small. This condition is stated in the

following corollary:

Corollary 6.3. If the conditions in (44) hold, then the unique coexistence equilib-

rium (N1, N2, P ) given by (49) is asymptotically stable in model (43) whenever:

d1 + g <

√Dd2(1−Dh) +

[d2Dh

2

]2− d2Dh

2. (47)

Proof. We know that if the conditions (44) hold, then the unique coexistence equi-

librium (N1, N2, P ) given by (49) in model (43) is asymptotically stable whenever

D(d1 + g)(1−Dh) > d2(d1Dh+ gDh+ d2). So, let:

51

Dd2(1−Dh) > (d1 + g)(d1 + g + d2Dh)

⇒ Dd2(1−Dh) > (d1 + g)2 +Dhd2(d1 + g)

⇒ Dd2(1−Dh) +

[Dhd22

]2> (d1 + g)2 +Dhd2(d1 + g) +

[Dhd22

]2⇒ Dd2(1−Dh) +

[Dhd22

]2> ((d1 + g) +

Dhd22

)2

⇒

√Dd2(1−Dh) +

[Dhd22

]2> (d1 + g) +

Dhd22

⇒ d1 + g <

√Dd2(1−Dh) +

[Dhd22

]2− d2Dh

2.�

(48)

We can also find a necessary condition for asymptotic stability for this this special

case. As we did in the first special case, we will also make use of Lemma 6.3 here.

Corollary 6.4. h(D + d1 + g) < 1 is a necessary condition for asymptotic stability

of the coexistence equilibrium N1, N2, P ) given by (49) in model (43).

Proof. Applying the results of Corollary 6.3 and Lemma 6.3, we have:

d1 + g <Dd2(1−Dh)

2(Dhd22

)

⇒d1 + g <1−Dh

h

⇒(d1 + g)h < 1−Dh

⇒h(D + d1 + g) < 1.�

(49)

We also give an example of coexistence in model (43) with numerical simulations.

Given the following set of ecological parameters: {b2 = 0.8, g = 0.2, d1 = 0.3, d2 =

0.2, e = 1.0, s = 0.3, h = 0.5, D = 0.1}, the condition for asymptotic stability of

the coexistence equilibrium is ensured by Theorem 6.5. The coexistence equilibrium

52

given by (49) is

(N1, N2, P ) = (0.5442176872, 0.3401360544, 0.4081632654).

With initial populations N1(0) = 0.2, N2(0) = 0.3, P (0) = 0.1, the numerical

simulation of model (43) demonstrated in Figure 4 shows that all three populations

converge to the equilibrium values for coexistence.

Figure 4: Asymptotic Stability, Coexistence While Only Adults Vulnerable

53

7 COEXISTENCE WHILE JUVENILE AND ADULT PREY ARE BOTH

VULNERABLE

The past two models represent extreme cases in model (1). In general, it is

more likely that a predator will consume both the juvenile prey and the adult

prey. In addition, the stage-structure in model (1) does play an important role

in most predator-prey relationships. Different population densities, physical sizes,

and predator handling times for juvenile and adult prey directly affect the dynam-

ics in model (1). Assuming that both juvenile and adult prey are vulnerable to the

predator, we explore the nature of the coexistence state and its stability in model (1),

with distinct capture rates, handling times and nutritional values which correspond

to juvenile and adult prey.

Theorem 7.1. If the conditions Dh1 < 1, Dh2 < 1, and b2g > d2(d1+g) hold, then

the stage-structured predator-prey model (1) has a unique coexistence equilibrium

(N1, N2, P ) given in (64), where r is the adult-to-juvenile ratio in the prey population

given in (60).

Proof.

The coexistence equilibrium in model (1) can be solved from the following system

of algebraic equations:

b2N2 − d1N1 − gN1 −s1N1P

1 + e1s1h1N1 + e2s2h2N2

= 0

gN1 − d2N2 −s2N2P

1 + e1s1h1N1 + e2s2h2N2

= 0

P (e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D) = 0

(50)

54

First consider the first equation in the system:

b2N2 − d1N1 − gN1 −s1N1P

1 + e1s1h1N1 + e2s2h2N2

= 0

⇒ b2N2 − d1N1 − gN1 =s1N1P

1 + e1s1h1N1 + e2s2h2N2

⇒ b2N2 − d1N1 − gN1

s1N1

=P

1 + e1s1h1N1 + e2s2h2N2

(51)

Now, consider the second equation in the system:

gN1 − d2N2 −s2N2P

1 + e1s1h1N1 + e2s2h2N2

= 0

⇒ gN1 − d2N2 =s2N2P

1 + e1s1h1N1 + e2s2h2N2

⇒ gN1 − d2N2

s2N2

=P

1 + e1s1h1N1 + e2s2h2N2

(52)

Notice that we obtain two equations whose right-hand side is P1+e1s1h1N1+e2s2h2N2

.

Therefore, the left-hand side of each of these two equations must be equal, that is:

b2N2 − d1N1 − gN1

s1N1

=gN1 − d2N2

s2N2

(53)

Using (58), we can obtain the adult-to-juvenile ratio r = N2/N1in the prey popula-

tion as follows:

b2N2 − d1N1 − gN1

s1N1

=gN1 − d2N2

s2N2

⇒ b2s2N22 − d1s2N1N2 − gs2N1N2 = gs1N

21 − d2s1N1N2

⇒ b2s2N22 + (d2s1 − d1s2 − gs2)N1N2 = gs1N

21

⇒ b2s2N2

N1

2

+ (d2s1 − d1s2 − gs2)N2

N1

− gs1

⇒ N2

N1

=d1s2 + gs2 − d2s1 ±

√(d1s2 + gs2 − d2s1)2 + 4gs1s2b22b2s2

(54)

It appears as though there are two solutions for the adult-to-juvenile ratio. However,

55

one of these solutions will always be negative. So, we will only choose the positive

one, that is:

r =d1s2 + gs2 − d2s1 +

√(d1s2 + gs2 − d2s1)2 + 4gs1s2b22b2s2

(55)

Substituting N2 = rN1 into the third equation in (55), we can then solve for N1:

P (e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D) = 0

⇒ e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D = 0

⇒ e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

= D

⇒ e1s1N1 + e2s2rN1

1 + e1s1h1N1 + e2s2h2rN1

= D

⇒ e1s1N1 + e2s2rN1 = D +De1s1h1N1 +De2s2h2rN1

⇒ e1s1N1 + e2s2rN1 −De1s1h1N1 −De2s2h2rN1 = D

⇒ N1(e1s1 + e2s2r −De1s1h1 −De2s2h2r) = D

⇒ N1(e1s1(1−Dh1) + e2s2r(1−Dh2)) = D

⇒ N1 =D

e1s1(1−Dh1) + re2s2(1−Dh2)

(56)

In order for N1 to be positive, it is sufficient to have 1−Dh1 > 0 and 1−Dh2 > 0.

Since N2 = rN1 and r > 0 those conditions will also be sufficient for N2 > 0 Finally,

substituting our equilibrium values for N1 and N2 in to the second equation of (55),

56

we can solve for P :

gN1 − d2N2 −s2N2P

1 + e1s1h1N1 + e2s2h2N2

= 0

⇒ gN1 − d2rN1 −s2rN1P

1 + e1s1h1N1 + e2s2h2rN1

= 0

⇒ (g − d2r)D

e1s1(1−Dh1) + re2s2(1−Dh2)−

s2rD

e1s1(1−Dh1) + re2s2(1−Dh2)P

1 +(e1s1h1 + e2s2h2r)D

e1s1(1−Dh1) + re2s2(1−Dh2)

= 0

⇒ (g − rd2)D

e1s1(1−Dh1) + re2s2(1−Dh2)− s2rDP

e1s1 + re2s2= 0

⇒ (g − rd2)D

e1s1(1−Dh1) + re2s2(1−Dh2)=

s2rDP

e1s1 + re2s2

⇒ P =(e1s1 + re2s2)(g − rd2)

rs2(e1s1(1−Dh1) + re2s2(1−Dh2))(57)

In order for P > 0, we must have that g − rd2 > 0 since we already have that

1−Dh1 > 0 and 1−Dh2 > 0. It is worth noting that we also could have solved for

P by substituting our values for N1 and N2 into the first equation of (55) as follows:

b2N2− d1N1 − gN1 −s1N1P

1 + e1s1h1N1 + e2s2h2N2

= 0

⇒ b2rN1 − d1N1 − gN1 −s1N1P

1 + e1s1h1N1 + e2s2h2rN1

= 0

⇒ (b2r − d1 − g)D

e1s1(1−Dh1) + re2s2(1−Dh2)−

s1D

e1s1(1−Dh1) + re2s2(1−Dh2)P

1 +(e1s1h1 + e2s2h2r)D

e1s1(1−Dh1) + re2s2(1−Dh2)

= 0

⇒ (b2r − d1 − g)D

e1s1(1−Dh1) + re2s2(1−Dh2)− s1DP

e1s1 + re2s2= 0

⇒ (b2r − d1 − g)D

e1s1(1−Dh1) + re2s2(1−Dh2)=

s1DP

e1s1 + re2s2

⇒ P =(e1s1 + re2s2)(b2r − d1 − g)

s1(e1s1(1−Dh1) + re2s2(1−Dh2))(58)

We also see that for P > 0, b2r − d1 − g > 0 Hence, the population densities in the

57

coexistence equilibrium (N1, N2, P ) of (1) are:

N1 =D

e1s1(1−Dh1) + re2s2(1−Dh2)

N2 =rD

e1s1(1−Dh1) + re2s2(1−Dh2)

P =(e1s1 + re2s2)(g − rd2)

rs2(e1s1(1−Dh1) + re2s2(1−Dh2))

(59)

It is obvious that r, N1, N2 > 0, and P > 0 if g− rd2 > 0 which yields the following

lemma:

Lemma 7.2. g − rd2 > 0 iff b2g > d2(d1 + g)

Proof. Let g − rd2 > 0

58

⇔ g −d1s2 + gs2 − d2s1 +

√(d1s2 + gs2 − d2s1)2 + 4gs1s2b22b2s2

d2 > 0

⇔ g >d1s2 + gs2 − d2s1 +

√(d1s2 + gs2 − d2s1)2 + 4gs1s2b22b2s2

d2

⇔ 2gb2s2 > d2(d1s2 + gs2 − d2s1 +√(d1s2 + gs2 − d2s1)2 + 4gs1s2b2)

⇔ 2gb2s2 − d2(d1s2 + gs2 − d2s1) > d2√(d1s2 + gs2 − d2s1)2 + 4gs1s2b2

⇔ [2gb2s2 − d2(d1s2 + gs2 − d2s1)]2 > d22[(d1s2 + gs2 − d2s1)

2 + 4gs1s2b2]

⇔ [2gb2s2 − d2(d1s2 + gs2 − d2s1)]2 − d22[(d1s2 + gs2 − d2s1)

2 + 4gs1s2b2] > 0

⇔ 4g2b22s22 − 4gb2s2d2(d1s2 + gs2 − d2s1) + d22(d1s2 + gs2 − d2s1)

2

− d22[(d1s2 + gs2 − d2s1)2 + 4gs1s2b2] > 0

⇔ 4g2b22s22 − 4gb2s2d2(d1s2 + gs2 − d2s1)− 4d22gs1s2b2 > 0

⇔ 4gb2s2[gb2s2 − d2(d1s2 + gs2 − d2s1)− d22s1] > 0

⇔ 4gb2s2[gb2s2 − d1d2s2 − d2gs2 + d22s1 − d22s1] > 0

⇔ 4gb2s22(gb2 − d1d2 − d2g) > 0

⇔ gb2 − d1d2 − d2g > 0

⇔ b2g > d2(d1 + g).�(60)

This anaylsis also concludes the details of the proof of the theorem. �

Let us consider our conditions that involve r. We have that g − rd2 > 0 and

b2r−d1−g > 0. The former implies that r < gd2

and the latter implies that r > d1+gb2

.

Putting these together we have that:

d1 + g

b2< r <

g

d2⇒ d2(d1 + g) < r < b2g (61)

So, we see that the adult-to-juvenile ratio is bounded below and above.

59

Theorem 7.3. If the assumptions of Theorem 7.1 hold, then the coexistence equi-

librium for model (1) given in (60) and (64) is

• asymptotically stable when A1A2 > A3,

• unstable when A1A2 < A3,

where:

A1 =1

rs2(e1s1 + re2s2){e2s22r[r(d2Dh2 + d1 + g) + g(1−Dh2)]

+ s1[e1s1(1−Dh1) + e2s2r](g − rd2) + e1s1s2(gr + d1r + g)};

A2 =A2 −A2

s2r2(e1s1 + re2s2)where

A2 = s1(g − rd2)[e1s1(g + rD)(1−Dh1) + gre2s2(1−Dh2)]

+ e2s22r

2(g + d1 +D)[rd2Dh2 + g(1−Dh2)] + e1s1s2rg(d1 + g),

A2 = s2r2[b2g(e1s1 + re2s2) + re2s2] + s1s2rD(g − rd2)[rb2e1h1 + e2h2(g − rd2)];

A3 =1

s2r2(e1s1 + re2s2){D(g − rd2)[e2s

22r

2(g + d1)(1−Dh2)

+ e1s1(s1g + s2b2r2)(1−Dh1) + e2s1s2r(1−Dh2)(2g − rd2)]}.

(62)

.

Proof.

Let:

y1 =dN1

dt

y2 =dN2

dt

y3 =dP

dt

(63)

60

The Jacobian matrix for this equilibrium in model (1)is:

J(N1, N2, P ) =

dy1dN1

dy1dN2

dy1dP

dy2dN1

dy2dN2

dy2dP

dy3dN1

dy3dN2

dy3dP

where

dy1dN1

=∂

∂N1

(b2N2 − d1N1 − gN1 −s1N1

1 + e1s1h1N1 + e2s2h2N2

)

= −d1 − g − s1P (1 + e2s2h2N2)

(1 + e1s1h1N1 + e2s2h2N2)2

dy1dN2

=∂

∂N2

(b2N2 − d1N1 − gN1 −s1N1

1 + e1s1h1N1 + e2s2h2N2

)

= b2 +e2s1s2h2N1P

(1 + e1s1h1N1 + e2s2h2N2)2

dy1dP

=∂

∂P(b2N2 − d1N1 − gN1 −

s1N1

1 + e1s1h1N1 + e2s2h2N2

)

= − s1N1

1 + e1s1h1N1 + e2s2h2N2

dy2dN1

=∂

∂N1

(gN1 − d2N2 −s2N2

1 + e1s1h1N1 + e2s2h2N2

)

= g +e1s1s2h1N2P

(1 + e1s1h1N1 + e2s2h2N2)2

dy2dN2

=∂

∂N2

(gN1 − d2N2 −s2N2

1 + e1s1h1N1 + e2s2h2N2

)

= −d2 −s2P (1 + e1s1h1N1)

(1 + e1s1h1N1 + e2s2h2N2)2

dy2dP

=∂

∂P(gN1 − d2N2 −

s2N2

1 + e1s1h1N1 + e2s2h2N2

)

= − s2N2

1 + e1s1h1N1 + e2s2h2N2

dy3dN1

=∂

∂N1

(P (e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D))

=e1s1P [1 + e2s2(h2 − h1)N2]

(1 + e1s1h1N1 + e2s2h2N2)2

dy3dN2

=∂

∂N2

(P (e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D))

61

=e2s2P [1 + e1s1(h1 − h2)N2]

(1 + e1s1h1N1 + e2s2h2N2)2

dy3dP

=∂

∂P(P (

e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D))

=e1s1N1 + e2s2N2

1 + e1s1h1N1 + e2s2h2N2

−D

and N1, N2 and P are the equilibrium values given in (64). After substitut-

ing the coexistence equilibrium (64) into the Jacobian matrix,the determinant of

(J(N1, N2, P )− λI) is: λ3 + A1λ2 + A2λ+ A3 where

A1 =1

rs2(e1s1 + re2s2){e2s22r[r(d2Dh2 + d1 + g) + g(1−Dh2)]

+ s1[e1s1(1−Dh1) + e2s2r](g − rd2) + e1s1s2(gr + d1r + g)};

A2 =A2 −A2

s2r2(e1s1 + re2s2)where

A2 = s1(g − rd2)[e1s1(g + rD)(1−Dh1) + gre2s2(1−Dh2)]

+ e2s22r

2(g + d1 +D)[rd2Dh2 + g(1−Dh2)] + e1s1s2rg(d1 + g),

A2 = s2r2[b2g(e1s1 + re2s2) + re2s2] + s1s2rD(g − rd2)[rb2e1h1 + e2h2(g − rd2)];

A3 =1

s2r2(e1s1 + re2s2){D(g − rd2)[e2s

22r

2(g + d1)(1−Dh2)

+ e1s1(s1g + s2b2r2)(1−Dh1) + e2s1s2r(1−Dh2)(2g − rd2)]}.

(64)

Here r is the adult-to-juvenile ratio in the coexistence equilibrium given by (64).

With the assumptions of Theorem 7.1, we already know that Dh1, Dh2 < 1 and

g − rd2 > 0. One can see that A1, A3 > 0. By the Routh-Hurwitz criteria for

stability, the coexistence equilibrium in (16) is asymptotically stable if and only if

A1A2 > A3. �

We will now illustrate, with a couple examples, the dynamics of the populations

in model (1) with respect to the presence and stability of the coexistence equilibrium

62

stated in Theorem 7.1 and Theorem 7.3.

First consider the following set of parameters: {b2 = 0.8, g = 0.2, d1 = 0.1, d2 =

0.2, e1 = 0.4, e2 = 0.8, s1 = 0.8, s2 = 0.4, h1 = 0.1, h2 = 0.2, D = 0.2}, which

satisfies the assumptions of Theorem 7.1. The adult-to-juvenile ratio given in (60)

is r = 0.6473635430 and the coexistence equilibrium given by (64) is

(N1, N2, P ) = (0.3902666860, 0.2526444246, 0.2801688772).

These ecological parameters make A1 = 0.8224784350 > 0, A3 = 0.02405818742 > 0,

and A1A2 − A3 = 0.00118934136 > 0. Theorem 57.3 indicates that the coexistence

equilibrium is asymptotically stable. The corresponding Jacobian Matrix is:

J(N1, N2, P ) =

−0.5152455069 0.8052906557 −0.3035152757

0.2008562444 −0.3072329285 −0.09824236215

0.08541320718 0.08367008052 0

The resulting eigenvalues are:

• λ1 = −0.8207899725,

• λ2 = −0.0008442314 + 0.1712025191 ∗ i

• λ3 = −0.0008442314− 0.1712025191 ∗ i

Since each of these eigenvalues have negative real part, then the coexistence equi-

librium is indeed asymptotically stable. Figure 5 demonstrates the numerical sim-

ulation of model (1) with initial populations N1(0) = 0.2, N2(0) = 0.3, P (0) = 0.1.

Next, we consider another set of parameters: {b2 = 0.8, g = 0.2, d1 = 0.3, d2 =

0.2, e1 = 0.5, e2 = 1.0, s1 = 0.6, s2 = 0.3, h1 = 0.5, h2 = 0.8, D = 0.2}. These

ecological parameters also satisfy the assumptions of Theorem 7.1, hence model (1)

63

Figure 5: Asymptotic Stability, Coexistence While Both Juvenile and Adults Vul-nerable

has a unique coexistence equilibrium. The adult-to-juvenile ratio given in (60) is

r = 0.7723635430 and the coexistence equilibrium given by (64) is

(N1, N2, P ) = (0.4304448356, 0.3324598983, 0.2248486898).

These ecological parameters make A1 = 0.8660746612 > 0, A3 = 0.01170074000 > 0,

and A1A2− A3 = −0.0049081529 < 0. Theorem 7.3 indicates that the coexistence

equilibrium is unstable. The corresponding Jacobian Matrix is:

J(N1, N2, P ) =

−0.6112392177 0.8106425872 −0.225687333

0.2025687332 −0.2548354442 −0.08715633380

0.0530508756 0.04951415056 0

The resulting eigenvalues are:

• λ1 = −0.8724569750,

• λ2 = 0.00319115654 + 0.1157629825 ∗ i

64

• λ3 = 0.00319115654− 0.1157629825 ∗ i

Since λ1 < 0, λ2 and λ3 have positive real parts, then the coexistence equilibrium is

an unstable saddle point. Figure 6 demonstrates the numerical simulation of model

(1) with initial populations N1(0) = 0.2, N2(0) = 0.3, P (0) = 0.1.

Figure 6: Instability, Coexistence with Cyclic Behavior

65

8 DISCUSSION/CONCLUSION

In this paper, we work on mathematical analysis of the predator-prey model

(1) with a stage-structured prey, which was initially presented and studied in [1].

This model has three possible types of equilibrium points: (0, 0, 0), (N1, N2, 0), and

(N1, N2, P ), representing different ecological outcomes.

The asymptotic stability of the trivial equilibrium is ensured by the condition

b2g < d2(d1 + g) (Theorem 5.1). Both the predator and prey will go to extinction if

the difference between the birth-to-death ratio b2/d2 of the adult prey and the death-

to-transition ratio d1/g of the juvenile prey is less than 1. When the difference is

exactly 1, the system has infinitely many prey-only equilibria (N1, N2, 0) with adult-

to-juvenile ratio as g/d2. The condition in Theorem 5.3, which implies the stability of

each prey-only equilibrium, indicates that the survival of the prey and extinction of

the predator can be simply ensured if the death rate D of the predator is sufficiently

large or the capture rate (si) and relative nutritional value (ei) of prey are sufficiently

small. The multiplicity of equilibria were demonstrated in Fig. 1 and Fig. 2.

The main focus of the analysis in this paper is to examine the presence and

asymptotic stability of a unique coexistence equilibrium, while the difference between

the ratios b2/d2 and d1/g is larger than 1 and the product of predator death rate

and handling time Dhi < 1. To illustrate the importance of our stage-structured

model, we first considered the asymptotic stability of the coexistence equilibrium of

our control model (11). We saw that when we do not consider age-structure, the

critical point of the coexistence equilibrium is always unstable. We then looked at

our age-structured model. For special cases where only one stage (adult or juvenile)

of the prey is vulnerable, we see that the equilibrium values of the species in (36) and

(49) depend on all ecological parameters, but the stability conditions (in Theorems

6.2 and 6.5) are not affected by the capture rate si and relative nutritional value ei of

66

the prey. The prey and predator will coexist when the death rate of adult d2 or the

sum of prey death rate and transition rate d1+g is small enough. These conclusions

are similar to the results in [1], under some simplified condition for parameters.

When adult and juvenile prey are both vulnerable, we find the results on exact

values of the unique coexistence equilibrium and its asymptotic stability. These

results are made possible by the discovery of the adult-to-juvenile ratio r given in

(60). The preliminary conditions given by Theorem 7.11 lead to the estimation

that r > (d1 + g)/b2, which means that the adult prey has to be relatively sizable

comparing to the juvenile prey. It is also seen here that smaller birth rate b2 of the

adult prey will actually raise the adult-to-juvenile ratio in the coexistence equilibrium

of the model. The complexity of the model is demonstrated in both the equilibrium

values (64) and the condition for asymptotic stability (Theorem 7.3). In this case,

each of the ecological parameters (including si and ei) plays its unique role in the

dynamics and equilibrium of the model, and should not be ignored. The stability

condition of the coexistence equilibrium obtained in Theorem 7.3 is demonstrated

through simulations in Fig. 5. When the condition fails, we observe in Fig. 6 a

different type of dynamics for long-term coexistence of all the species with cyclic

pattern and increasing population sizes.

In conclusion, we see that the age-structured aspect of our model enables the

system to have stability of a coexistence equilibrium which otherwise would not be

possible.

67

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BIOGRAPHICAL SKETCH

I, Michael Cowen, am originally from Winston-Salem, NC where I attended R.J.

Reynolds High School. I started my college career in 2008 at North Carolina State

University where I eventually obtained a B.S. in Applied Mathematics and a minor

in Spanish. Upon graduating from North Carolina State University I was presented

with the opportunity to continue my study of Mathematics at the University of North

Carolina-Wilmington. While obtaining my M.S. at UNCW, I have worked as a tutor

in the Math Lab, been a grader for a variety of Calculus classes and have taught my

own sections of College Algebra. During one of the summers of my graduate career,

I also helped organize and plan the 9th annual AIMS Conference which was held in

Orlando, Fl. In the Fall, I will be enrolling into a Doctoral Program in Mathematics

at the University of South Carolina.

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