n-Dimensional ratio-dependent predator–prey systems with diffusion

Applied Mathematics and Computation 205 (2008) 325–335

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

n-Dimensional ratio-dependent predator–prey systems with diffusion q

Krisztina KissInstitute of Mathematics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary

a r t i c l e i n f o

Keywords:Predator–prey systemFunctional responseSign stabilityRatio dependenceDiffusionTuring instability

0096-3003/$ - see front matter � 2008 Elsevier Incdoi:10.1016/j.amc.2008.08.030

q The present work has partially been supported bE-mail address: [email protected]

a b s t r a c t

This paper deals with a ratio-dependent predator–prey system with diffusion. We willinvestigate under what conditions Turing stability or instability occurs in higherdimensions.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

Let us consider a ratio-dependent ecological system in which n different predator species (the ith predator quantities attime t are denoted by yiðtÞ, i ¼ 1;2; . . . ; n; respectively) are competing for a single prey species (the quantity of prey at time tis denoted by xðtÞ):

_x ¼ rxgðx;KÞ �Pni¼1

yipiyix

� �;

_yi ¼ yipiyix

� �� diyi; i ¼ 1;2; . . . ;n;

9>=>; ð1Þ

where dot means differentiation with respect to time t. We assume that the per capita growth rate of prey in absence of pre-dators is rgðx;KÞ where r is a positive constant (in fact the maximal growth rate of prey), K > 0 is the carrying capacity ofenvironment with respect to the prey. The function g satisfies some natural conditions, see the details in [7,4], namelyg 2 C2ðð0;1Þ � ð0;1Þ;RÞ, g 2 C0ð½0;1Þ� ð0;1Þ;RÞ,

gð0;KÞ ¼ 1; g0xðx;KÞ < 0 < g00xKðx;KÞ ðx > 0;K > 0Þ; ð2ÞlimK!1

g0ðx;KÞ ¼ 0 ð3Þ

uniformly in ½d; x0� for any 0 < d < x0, and the (possibly) improper integralR x0

0 g0xðx;KÞdx is uniformly convergent in ½K0;1Þfor any K0 > 0,

ðK � xÞgðx;KÞ > 0 ðx P 0; x–K > 0Þ: ð4Þ

A possible choice of function g satisfying conditions (2)–(4) may be the so-called logistic growth rate of prey

gðx;KÞ ¼ 1� xK: ð5Þ

. All rights reserved.

y the National Scientific Research Foundation Nos. T 047132 and NK 63066.

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326 K. Kiss / Applied Mathematics and Computation 205 (2008) 205 (2008) 325–335

We assume further that the death rate di > 0 of predator i is constant and the per capita birth rate of the same predator ispi

yix

� �where function pi also satisfies some natural conditions, see also in [7], namely pi 2 C1ðð0;1Þ � ð0;1Þ � ð0;1Þ;RÞ,

pi 2 C0ð½0;1Þ� ð0;1Þ� ð0;1Þ;RÞ,

pið0; yi; aiÞ ¼ 0; p0ixðx; yi; aiÞ > 0 ðx > 0; yi > 0; ai > 0Þ; ð6Þ

p0ixðx; yi; aiÞ <piðx; yi; aiÞ

xðx > 0; yi > 0; ai > 0Þ; ð7Þ

p0iaiðx; yi; aiÞ 6 0 ðx > 0; yi > 0; ai > 0Þ: ð8Þ

In paper [7] we have already investigated the system with the so-called Michaelis–Menten or Holling type functional re-sponse (see in [9]) in case of ratio-dependence:

piyi

x; ai

� �¼ mi

1ai

yix þ 1

¼ mix

aiyi þ x: ð9Þ

In case of a biological model this function is called a Holling type function and usually in case of a biochemical model it iscalled Michaelis–Menten type function without ratio-dependence. In paper [7] we have also investigated the system withthe Ivlev type functional response (see in [9]) in case of ratio-dependence:

piyi

x; ai

� �¼ mi 1� e�

xai yi

� �; ð10Þ

where parameter ai is the so-called ‘‘half-saturation constant”, namely in the case where pi is a bounded function for fixedai > 0, mi ¼ supx;yi>0piðx; yi; aiÞ is the ‘‘maximal birth rate” of the ith predator. (To save space we do not write out the depen-dence on ai in (1).) For the survival of predator i it is, clearly, necessary that the maximal birth rate be larger than the deathrate:

mi > di: ð11Þ

This will be assumed in the sequel. Finally, we assume that the presence of predators decreases the growth rate of prey bythe amount equal to the birth rate of the respective predator.

2. Model with diffusion

Let us modify the system (1) assuming that prey and predators are diffusing according to Fick’s law (that is, in the direc-tion of the negative gradient of the density and with a speed that is proportional to the modulus of the gradient) in the inter-val n 2 ½0; l�, i.e. let us consider the reaction–diffusion system

x0t ¼ dxx00nn þ rxgðx;KÞ �Pni¼1

yipiyix

� �;

y0it ¼ dyiy00i nn þ yipi

yix

� �� diyi; i ¼ 1;2; . . . ; n

9>=>; ð12Þ

on ðt; nÞ 2 ð0;1Þ� ð0; lÞ; l > 0. We are interested in solutions x; yi : ½0;1Þ � ½0; l� ! ½0;1Þ, i ¼ 1;2; . . . ;n that satisfy the no-flux boundary conditions:

x0nðt; 0Þ ¼ x0nðt; lÞ ¼ y0inðt; 0Þ ¼ y0inðt; lÞ ¼ 0 ðt 2 ð0;1ÞÞ; i ¼ 1;2; . . . ;n ð13Þ

and the initial conditions

xð0; nÞ ¼ x0ðnÞ; yið0; nÞ ¼ yi0ðnÞ; ðn 2 ½0; l�Þ:

The diffusion coefficients dx; dyi, i ¼ 1;2; . . . ;n are supposed to be positive. This modification is the same as done by Cavani

and Farkas [2] in two dimensions. That system was a two-dimensional one only, and they used a special Holling type func-tional response without ratio-dependence and a logistic growth rate of the prey, but the mortality of predator was not equalto a constant. There is a generalization of this model in case of two-dimensional spatial coordinate by Kovács [8]. In [8] theresults of [2] are extended concerning intervals to certain two-dimensional planar domains (hexagon, rectangle, triangle)where explicit conditions are established under which the reaction–diffusion system considered undergoes a Turing bifur-cation. We are going to investigate the n-dimensional system in case of general ratio-dependent functional responses withconstant death rates of predators on a one-dimensional spatial interval.

It is easy to see that the positive orthant is positively invariant. The proof of dissipativeness is more complicated. Theauthor with Lizana and Duque have already proven the dissipativeness see in [3] in case of a Michaelis–Menten type func-tional response. This proof also can be apply in case of an Ivlev type functional response. Thus, the model is biologically wellposed.

K. Kiss / Applied Mathematics and Computation 205 (2008) 325–335 327

Introducing the ðnþ 1Þ-dimensional vector U ¼ colðx; y1; . . . ; ynÞ, the diagonal matrix D ¼ diagðdx; dy1; . . . ; dyn

Þ and thevector

FðUÞ ¼

rxgðx;KÞ �Pni¼1

yipiyix

� �y1p1

y1x

� �� d1y1

..

.

ynpnynx

� �� dnyn

266666664

377777775 ð14Þ

system (12) and boundary conditions (13) assume the form

U0t ¼ DU00nn þ FðUÞ; ð15ÞU0nðt;0Þ ¼ U0nðt; lÞ ¼ 0: ð16Þ

Clearly, a spatially constant solution UðtÞ ¼ colðxðtÞ; y1ðtÞ; . . . ; ynðtÞÞ of (15) satisfies the boundary conditions (16) and the ki-netic system

U0t ¼ FðUÞ; ð17Þ

which is, in fact, system (1). The equilibrium of (17) is a constant solution of (15) at the same time. We shall be concernedwith the positive equilibrium whose existence has been supposed in paper [7], namely it has been supposed that there existsan equilibrium point E�ðx�; y�1; . . . ; y�nÞ in the positive orthant, where x�, and y�i are the solutions of the following equations:

rxgðx;KÞ ¼Xn

i¼1

diyi; piyi

x

� �¼ di; i ¼ 1; . . . ;n: ð18Þ

Note, that x� > 0 if and only if K > x�.Now we follow the method suggested in [2] and we complete the direct generalization of its result using our previous

result published in [7].We linearize system (15) around point E�. Introducing the new coordinates V ¼ ðx� x�; y1 � y�1; . . . ; yn � y�nÞ the linearized

system assumes the form

V 0t ¼ DV 00nn þ AV ; ð19ÞV 0nðt;0Þ ¼ V 0nðt; lÞ ¼ 0; ð20Þ

where A is the same as given in [7], namely:

A ¼

a11 �d1 � y�1p01y�1x�

� �1x� . . . . . . �dn � y�np0n

y�nx�

� �1x�

y�1p01y�1x�

� �� y�1

x�2

� �y�1p01

y�1x�

� �1x� 0 . . . 0

..

. ... ..

. . .. ..

.

y�np0ny�nx�

� �� y�n

x�2

� �0 . . . 0 y�np0n

y�nx�

� �1x�

266666664

377777775; ð21Þ

where

a11 ¼ rgðx�;KÞ þ rx�g0xðx�;KÞ �Xn

i¼1

y�i p0iy�ix�

� �� y�i

x�2

� �ð22Þ

and p0iyix

� �¼ dpi

yixð Þ

dyixð Þ

.

We solve (19) and (20) linear boundary value problem by Fourier’s method. Solutions are looked for in the formVðt; nÞ ¼ UðtÞWðnÞ. The functions U : ½0;1Þ ! Rnþ1, W : ½0; l� ! R are to satisfy

_U ¼ ðA� a2DÞU; ð23Þ

where dot denotes differentiation with respect to time t, and

W00 ¼ �a2W; W0ð0Þ ¼ W0ðlÞ ¼ 0; ð24Þ

where prime denotes differentiation with respect to the spatial variable n. The eigenvalues of the boundary value problem(24) are

a2j ¼

jpl

� �2

; j ¼ 0;1;2; . . . ð25Þ


with corresponding eigenfunctions

WjðnÞ ¼ cosjpn

l

� �: ð26Þ

Clearly, 0 < a1 < a2 < � � � These eigenvalues are to be substituted into (23). Denoting ðnþ 1Þ independent solutions of (23)taken with a ¼ aj by U1j; . . . ;Unþ1j, the solution of the boundary value problem (19), (20) is obtained in the form

Vðt; nÞ ¼X1j¼0

Xnþ1

i¼1

cijUij

!cos

jpnl

� �; ð27Þ

where cij (i ¼ 1;2; . . . ;nþ 1, j ¼ 0;1;2; . . .) are to be determined according to the initial value Vð0; nÞ. If e.g. Uð0Þ ¼ I (I is theðnþ 1Þ � ðnþ 1Þ identity matrix), then

c10

c20

..

.

cnþ1 0

266664377775 ¼ 1

l

Z l

0Vð0; nÞdn; ð28Þ

c1j

c2j

..

.

cnþ1 j

266664377775 ¼ 2

l

Z l

0Vð0; nÞ cos

jpnl

� �dn ðj ¼ 1;2; . . .Þ: ð29Þ

The following notations will be used:

BðaÞ ¼ A� a2D; Bj ¼ BðajÞ ¼ A� a2j D: ð30Þ

Now we summarize some basic properties before stating our new results on stability behaviour.

2.1. Stability

An n� n matrix A ¼ ½aij� is stable if each eigenvalue has negative real part. The following definition can be found in [6]:

Definition 2.1. An n� n matrix A ¼ ½aij� is said to be sign-stable if each matrix eA of the same sign-pattern as A(sign~aij ¼ signaij for all i; j) is stable.

It is also known:

Definition 2.2. Given D, an n� n matrix A ¼ ½aij� is said to be strongly stable if A is stable and if for all j all the eigenvalues ofBj (given by (30)) have negative real parts. And matrix A ¼ ½aij� is excitable if it is stable and at least one eigenvalue of Bj haspositive real part.

Definition 2.3. We say that the equilibrium E� of (15) is Turing (diffusionally) unstable if it is an asymptotically stable equi-librium of the kinetic system (17) but it is unstable with respect to solutions of (15) and (16).

We have already proven the following in [7]:

Theorem 2.1. The matrix A defined by (21) is sign-stable and E� is an asymptotically stable equilibrium point of system (1) (orequivalently of system (17)) if

a11 6 0; ð31Þ

p0iy�ix�

� �< 0; i ¼ 1; . . . ;n ð32Þ

and

�di � y�i p0iy�ix�

� �1x�< 0; i ¼ 1; . . . ;n: ð33Þ

We can prove the following.

Theorem 2.2. If (31)–(33) are satisfied then the matrix A defined in (21) is strongly stable and the equilibrium E� of (15) isasymptotically stable.

Proof. It is enough to prove that (30) is sign-stable. We have to check the conditions of Theorem 2.6 of [6]. The entries at themain diagonal of (30) are negative. The product of the entries symmetrical to the main diagonal are trivially negative or zero.

1

2 3 n...

Fig. 1. The graph of matrix (30).

1

2 3 n...

Fig. 2. Colouring the vertices black whose belong to negative main diagonal entries, the graph of matrix (30).


The graph of (30) is a tree, see Fig. 1. (Note, vertex 1 can be also white.) Now we have to colour those vertices black whichbelong to negative main diagonal entries. We get the coloured graph, see Fig. 2. We can apply Theorem 2.6 of [6] on p. 13 andthe proof of theorem is complete. h

This theorem means that in case of a sign-stable interaction matrix Turing instability cannot occur. If a11 > 0 (given by(22)) then (21) is not sign-stable. This is the situation treated by Farkas and Cavani [2] in two dimension. That is the casewhen E� lies in the Allée-effect zone – here prey is scarce and an increase in prey quantity is beneficial for the growth rateof prey, see in [5]. To keep the prey growth rate zero the increase of prey must be counterbalanced by an increase of preda-tors. Let us consider this two-dimensional case first. Now function F given by (14) has two rows F1 and F2. Let us suppose thatany predator quantity growth will decrease the growth rate of prey, namely F 01y1

< 0. A typical reasonable form of the zeroisocline F1ðx; y1Þ ¼ 0, applicable to most species is shown in Fig. 3. and we can see that F 01x

> 0, thus a11 > 0 holds in the Allé-e-effect zone. (To complete these figures we used a Holling model. Similar situations occur in case of an Ivlev model.)

This meaning of the Allée-effect zone is also kept in higher dimension, namely in the Allée-effect zone an increase in preyquantity is beneficial for the growth rate of prey. In order to keep the prey growth rate zero the increase of the prey can becounterbalanced by the increase of the whole quantity of the different predators. Let us consider the higher dimensionalcases. Now function F given by (14) has nþ 1 rows Fi, i ¼ 1;2; . . . ; ðnþ 1Þ. Suppose that any predator’s quantity growth willdecrease the growth rate of prey, namely F 01yi

< 0, i ¼ 1;2; . . . ;n. In the three dimensional case a typical onion like preynullcline surface of F1ðx; y1; y2Þ ¼ 0 is shown in figure (see Fig. 2.4.2 in [5, p. 44]). Inside the onion like surface F1 > 0 whileoutside F1 < 0. Function F is increasing as we cross the surface inwards and therefore its gradient points inward. Therefore ifthe equilibrium point is on the eastern hemisphere of this onion then F 01x

< 0, thus, a11 < 0 and on the western hemisphere ofthe onion F 01x

> 0, thus, a11 > 0 and we can see that F 01x> 0, thus a11 > 0 in the Allée-effect zone.

Specially, if the system is a ratio-dependent one, some typical onion like surfaces are presented using Wolfram Mathem-atica 6.0 (http://www.wolfram.com) in Figs. 4–6. (To complete these figures we used a Holling model. Similar situations oc-cur in case of an Ivlev model. In case of r ¼ 3 the picture is more similar to an onion.) If r is low (see Fig. 4.) there is no positiveequilibrium point of the system. If r increases a nontrivial equilibrium appears on the surface denoted by a black sphere. (SeeFig. 5.) If we increase r further the surface changes according to the plain curves shown in picture 3. If r is big enough then the

0.02 0.04 0.06 0.08 0.10x

0.005

0.010

0.015

0.020

0.025

y1

m/a=r

m/a=r

m/a=r

0.00 0.02 0.04 0.06 0.08 0.10x

0.02

0.04

0.06

0.08

0.10

0.12

0.14

y1

0.02 0.04 0.06 0.08 0.10x

0.002

0.004

0.006

0.008

0.010

0.012

y1

Fig. 3. Typical nullclines of prey in case of two-dimension.

http://www.wolfram.com

0.00

0.05

0.10

x

0.000

0.005

0.010

y1

0.000

0.002

0.004

y2

Fig. 4. Typical zero cline of prey in case of r ¼ 3 in three dimensions (r ¼ 3; K ¼ 0:1; m1 ¼ 16; a1 ¼ 4; m2 ¼ 18; a2 ¼ 2).


surface ‘‘opens”. We can see that the equilibrium point moves on this surface as r increases. (The Mathematica notebooks tocreate the figures and also showing this movement of the equilibrium point can be found on the author’s homepage: http://math.bme.hu/~kk//ratiodiff#.) If it comes over the ‘‘ridge” then it leaves the Allée-effect zone. The equilibrium point is in theAllée-effect zone in Fig. 5, before the ‘‘ridge”, and it is outside of the Allée-effect zone in Fig. 6, after the ‘‘ridge”. (We note thatthe program Wolfram Mathematica 6.0 denotes the uncontinuity by the scallop of the surface near the origin or along to they1 (or y2) axis. If r is big enough then all points of the surface are outside of the Allée-effect zone.

If F 01yi< 0 (namely if yi is a predator of x) then a11 > 0 holds also in higher dimensions in the Allée-effect zone. To see this,

let us consider F1ðx; y1; . . . ; ynÞ ¼ rxgðx;KÞ �Pn

i¼1yipiyix

� �and the surface F1ðx; y1; . . . ; ynÞ ¼ 0, which is the prey nullcline sur-

face. Let E1 ¼ ðx1; y11; . . . ; y1

nÞ; E2 ¼ ðx2; y21; . . . ; y2

nÞ be two different points in the Allée-effect zone on the prey nullcline sur-face, where x1 < x2; y1

i < y2i ; i ¼ 1; . . . ;n

0 ¼ F1ðx2; y21; . . . ; y2

nÞ � F1ðx1; y11; . . . ; y1

nÞ¼ fF1ðx2; y2

1; . . . ; y2nÞ � F1ðx2; y1

1; y22 . . . ; y2

nÞg þ fF1ðx2; y11; y

22 . . . ; y2

nÞ � F1ðx2; y11; y

12; y

23; y

24; . . . ; y2

nÞgþ fF1ðx2; y1

1; y12; y

23; y

24; . . . ; y2

nÞ � F1ðx2; y11; y

12; y

13; y

24; . . . ; y2

nÞg þ . . .þ fF1ðx2; y11; y

12; y

13; . . . ; y1

n�1; y2nÞ

� F1ðx2; y11; y

12; y

13; . . . ; y1

n�1; y1nÞg þ fF1ðx2; y1

1; y12; y

13; . . . ; y1

nÞ � F1ðx1; y11; . . . ; y1

nÞg:

Expressions in the brackets are negative except the last bracket because of F 0yi< 0, thus Fx > 0 must hold.

It is reasonable to say that E� lies outside the Allée-effect zone if a11 < 0 and E� lies in the Allée-effect zone if a11 > 0.

Remark 2.1. Theorem 2.2 means that if E� lies outside the Allée-effect zone then Turing instability cannot occur. If E� isTuring unstable then it lies in the Allée-effect zone.

This remark is the direct generalization of Lemma 2.1 of [2].In the following section we will present our new results on instability behaviour.

2.2. Turing instability

2.2.1. Elementary calculationsBefore we present our result for the case of Turing instability we have to state some basic results connected with the

characteristic polynomial of a 3� 3 square matrix. Let us denote the matrix by A ¼ ½aij� again. Specially, let the sign of its

http://math.bme.hu/~kk//ratiodiff#

http://math.bme.hu/~kk//ratiodiff#

0.00

0.05

0.10

x

0.00

0.01

0.02

0.03y1

0.00

0.01

0.02

y2


0.00

0.05

0.10

x

0.00

0.01

0.02

0.03y1

0.00

0.02

0.04

0.06

0.08

y2




entries be the following: a11 > 0, a22; a33 < 0, a12; a13 < 0, a21; a31 > 0 and a23; a32 ¼ 0. Then, A has the following signpattern:

A ¼þ � �þ � 0þ 0 �

264375: ð34Þ

Let us denote the characteristic polynomial of A by

DðkÞ ¼ k3 þ a2k2 þ a1kþ a0 ð35Þ

with

a2 ¼ �a11 � a22 � a33; ð36Þa1 ¼ a11a22 þ a11a33 þ a22a33 � a12a21 � a13a31; ð37Þa0 ¼ �det A ¼ �a11a22a33 þ a22a13a31 þ a33a12a21: ð38Þ

It is known that the necessary condition of the stability of the characteristic polynomial is ai > 0; i ¼ 0;1;2; and the suffi-cient condition is obtained if a2a1 � a0 > 0 also holds, see e.g. in [5].

Lemma 2.1. Let A be a 3� 3 square matrix with the sign pattern given by (34). If a11 þ a22 < 0 and a11 þ a33 < 0 are satisfiedthen a2 > 0 in (36) and a2a1 � a0 > 0. Even if only

a11 þa22

2< 0 ð39Þ

and

a11 þa33

2< 0 ð40Þ

hold then a1 > 0 is also satisfied.

Proof. It is easy to see that a2 > 0. And

a2a1 � a0 ¼ ð�a22 � a33Þða11 þ a33Þða11 þ a22Þ þ a12a21ða11 þ a22Þ þ a13a31ða11 þ a33Þ > 0:

a1 ¼ a22 a11 þa33

2

� �þ a33 a11 þ

a22

2

� �� a12a21 � a13a31 > 0: �

Remark 2.2. Under conditions (39), (40) a0 can be either positive or negative. Some easy controllable conditions for a0 < 0are

12

a11a22 < a12a21 ð41Þ

and

12

a11a33 < a13a31: ð42Þ

If in these inequalities both we have > instead of < then a0 > 0.

Proof

a0 ¼ �a3312

a11a22 � a12a21

� �� a22

12

a11a33 � a13a31

� �: �

Instead of (41) and (42) we make other assumptions to control the sign of a0 in the special case of matrix (21) because thefollowing holds:

Theorem 2.3. Let E� be a positive equilibrium point of the kinetic system (17) in case of n ¼ 2. Let the matrix A be given by (21). Ifg0xðx�;KÞ < 0 and (32) holds then the constant term in its characteristic polynomial given by (38) is positive.

Proof. A short calculation shows that: a0 ¼ �det A ¼ y�1x� p01

y�1x�

� �y�2x� p02

y�2x�

� �ð�rx�g0xðx�;KÞÞ > 0: h

Condition g0ðx�;KÞ < 0 follows from a natural condition (2). This condition holds evidently if g is given by (5).

Lemma 2.2. Let C ¼ diagðc11; c22; c33Þ with cii > 0; i ¼ 2;3 and 0 < c11 < 1. Let us denote eA :¼ A� diagða11c11; a22c22; a33c33Þ,its characteristic polynomial by eDðkÞ ¼ k3 þfa2k2 þfa1kþfa0 . If A has the same sign pattern as (34) and satisfies (39) and (40) thenfa2 ;fa1 > 0 and fa2fa1 > fa0 .


Proof

eA ¼ a11ð1� c11Þ a12 a13

a21 a22ð1þ c22Þ 0

a31 0 a33ð1þ c33Þ

26643775: ð43Þ

Matrix eA has the same sign pattern as A and satisfies conditions (39) and (40) because a11ð1� c11Þ þ aii2 ¼ a11 þ aii

2

� ��

a11c11 < 0; i ¼ 1;2. Thus, because of Lemma 2.1 the proof is complete. h

Remark 2.3. The argumentation here is very similar to the one in [10] where the effect of diffusion on stable systems arestudied in three and n-species chemical reactions.

Now we can state our new result.

Theorem 2.4. Let A be a stable matrix with the same sign pattern as (34), satisfying conditions (39) and (40). If

a11l2

p2 > dx ð44Þ

is also satisfied then in case of an arbitrary a11 l2

p2 > dx > 0 there exist k 2 N and dy1; dy2

> 0 such that Bk (given by (30)) isunstable.

Proof. Let us fix dx according to condition (44). We will give dy1; dy2

> 0 such that B1 will be unstable. There exist0 < c11 < 1; c22; c33 > 0 and dy1 ; dy2 > 0 such that a2

1dx ¼ c11a11, a21dy1 ¼ �c22a22, a2

1dy2 ¼ �c33a33 hold. The constant c11 < 1because of (44). Thus, B1 ¼ eA (given in Lemma 2.2) satisfies conditions of Lemma 2.2, hence the characteristic polynomialof B1 is: DB1 ðkÞ ¼ k3 þ a2B1

k2 þ a1B1kþ a0B1

, such that a1B1; a2B1

> 0, a2B1a1B1

> a0B1where a0B1

¼ � 12 a33

� �ða11a22ð1� c11Þ

ð1þ c22Þ � 2a21a12Þð1þ c33Þ þ � 12 a22

� �ða11a33ð1� c11Þð1þ c33Þ � 2a31a13Þð1þ c22Þ < 0 if ða11a22ð1� c11Þð1þ c22Þ � 2a21a12Þ

< 0 and ða11a33ð1� c11Þð1þ c33Þ � 2a31a13Þ < 0. These last two inequalities are satisfied if

1þ c22 >2a12a21

a11a22ð1� c11Þ; ð45Þ

1þ c33 >2a13a31

a11a33ð1� c11Þð46Þ

are chosen. Now we can determine c22; c33 according to (45) and (46), and then dy1; dy2

can be calculated. h

Corollary 2.1. If A is stable with the same sign pattern as (34), satisfying conditions (39) and (40) and dx satisfies (44) then for alldx > 0 there exist dy1

; dy2such that A is stable without being strongly stable, i.e. it is excitable.

Remark 2.4. If matrix A has the same sign pattern as (34), it satisfies conditions (39) and (40) and the inequalities (41) and(42) are satisfied with the opposite direction then A is stable.

Remark 2.5. If A is stable with the same sign pattern as (34) and dx >a11 l2

p2 then for all dx; dy1; dy2

> 0 A is strongly stable.

Proof. In this case Bk is sign stable for all k > 0. h

It is a very interesting question how the diffusion coefficient dx (or dyi) can be changed. For example one of the major

problems is the destruction of the rainforests in South America. As a result, animals lose their home. If specially they aremixed food species, such as the prey species so the predators species have to change their diffusion coefficients, namely theyhave to move differently in order to survive. This destruction is also bringing about changes in the earth’s climate. There is abiological and physiological evidence (see for example in [1]), that when predators have to search for food and therefore haveto share or compete for food, a more suitable general predator prey theory should be based on the ratio-dependent theory.Our model also shows sensibility to the spatial movement of the species. In our model the spatial coordinate is one-dimen-sional, thus we can take into account also water pollution in a river. This can cause Turing instability too.

We note that condition (44) is not too strong because parameter l is the length of the space where the animals are dif-fusing, thus, it can be quite big.

Condition a11 > 0 means that the equilibrium point is outside the Allée-effect zone. Conditions (39), (40) mean that theintraspecific competition in the predator populations is strong enough. The advantage of the Theorem 2.4 is that its proofgives a method to determine the diffusion coefficients in order to guarantee Turing instability. This is denoted by the follow-ing example.


Example 2.1. Let us consider a three dimensional Holling type ratio-dependent model, namely g is given by (5) and pi isgiven by (9). Let the constants be given as follows: m1 ¼ 16;m2 ¼ 18; d1 ¼ 8; d2 ¼ 12; a1 ¼ 4; a2 ¼ 2; r ¼ 7: In this case theinteraction matrix of the diffusing animals is given by

A ¼1 �4 �81 �4 01 0 �4

264375: ð47Þ

Note a11 þ 12 a22 ¼ �1 < 0, a11 þ 1

2 a33 ¼ �1 < 0. The characteristic polynomial of A is DðkÞ ¼ ð�4� kÞðk2 þ 3kþ 12Þ. This is astable polynomial.

Let l ¼ p, dx ¼ 12, a1 ¼ p2

l2¼ 1, thus, c11 ¼ 1

2.2a12a21

a11a22ð1�c11Þ ¼ 4 < 1þ c22, thus, in case of c22 > 3, for example in case of c22 ¼ 4 we can have Turing instability.2a13a31

a11a33ð1�c11Þ ¼ 8 < 1þ c33, thus, in case of c33 > 7, for example in case of c33 ¼ 8 we will have Turing instability. Let us consider

e.g. the following matrix:

B1 ¼

12 �4 �81 �20 01 0 �36

264375: ð48Þ

Since det B1 ¼ 3296 > 0 hence B1 is unstable. The corresponding diffusion coefficients are: dy1¼ 16, dy2

¼ 32.The situation is modelled by Fig. 5, when the equilibrium point is inside the Allée-effect zone.

3. Discussion

System (12) describes the dynamics of an n predator–prey interaction. In absence of predation prey quantity grows by afunction g which can be for example given by (5). Predator mortality is constant. The functional response is of ratio-depen-dent type, which can be for example a Holling or an Ivlev type one (given by (9) and (10), respectively). Both species aresubject to Fickian diffusion in a one dimensional spatial habitat from which and into which there is no migration. It is as-sumed that the system has a positive equilibrium in the positive orthant. If this equilibrium is outside of the Allée-effectzone –, i.e. in a neighbourhood of the equilibrium the increase of prey density is not beneficial to prey’s growthrate – and the interaction matrix of the kinetic system is sign-stable then the equilibrium point remains locally asymptot-ically stable for any diffusion rate of species.

If this equilibrium is inside the Allée-effect zone, i.e. in a neighbourhood of the equilibrium the increase of prey density isbeneficial to prey’s growth rate and obviously the interaction matrix of the kinetic system is not sign-stable but it has a spe-cial sign-pattern then the equilibrium point remains locally asymptotically stable if the prey diffusion rate is relatively highcompared e.g. to the square of the length of the spatial domain for any diffusion rate of predator species.

If the prey diffusion rate is lower (condition (44)) then one may increase the predator diffusion rate to a value at whichequilibrium loses its stability. To be sure, the equilibrium point considered as an equilibrium of the system without diffusionstays stable, i.e. a so called diffusional instability occurs quite naturally, contrary to the general belief that diffusion usuallystabilize systems. See more examples in [11] We gave a method to determine the diffusion coefficient in order to have Tur-ing-instability.

These results correspond to the result of [2] proven in two dimensions and now most of those results are proven also inhigher dimensions. Importance of an Allée-effect zone is proven too.

Acknowledgements

This work is partly the generalization of a paper of Cavani and Farkas [2]. The author was a student of the late Prof. MiklósFarkas of Budapest University of Technology and Economics. They worked together for more than 20 years. Prof. Miklós Far-kas regrettably died on the 28th of August 2007. The author is honoured to have known him, and remembers him with greatfondness, love and gratitude. She is eternally thankful to him for his precious ideas and comments throughout so many years.

References

[1] H.R. Akcakaya, R. Arditi, L.R. Ginzburg, Ratio-dependent prediction: an abstraction that works, Ecology 76 (1995) 995–1004.[2] M. Cavani, M. Farkas, Bifurcations in a predator–prey model with memory and diffusion II: turing bifurcation, Acta Math. Hungar. 63 (4) (1994) 375–

393.[3] C. Duque, K. Kiss, M. Lizana, On the dynamics of an n-dimensional ratio-dependent predator–prey system with diffusion, Appl. Math. Comput.,

submitted for publication.[4] M. Farkas, Competitive Exclusion by Zip Bifurcation, IIASA Workshop, Sopron, Lecture Notes in Economics and Mathematical Systems, vol. 287,

Springer-Verlag, Berlin, 1987, pp. 165–178.[5] M. Farkas, Dynamical Models in Biology, Academic Press, New York, 2001.[6] C. Jeffries, V. Klee, P. van den Driessche, Qualitative stability of linear systems, Linear Algebra Appl. 87 (1987) 1–48.


[7] K. Kiss, S. Kovács, Qualitative behaviour of n-dimensional ratio-dependent predator–prey systems, Appl. Math. Comput. 199 (2) (2008)535–546.

[8] S. Kovács, Pattern formation in bounded spatial domains, Univ. Sci. Budapest. Eötvös Sect. Math. 42 (2000) 185–203.[9] R.M. May (Ed.), Theoretical Ecology, second ed., Blackwell, Oxford, 1981.

[10] L. Szili, J. Tóth, Necessary condition of the Turing instability, Phys. Rev. E 48 (1) (1993) 183–186.[11] L. Szili, J. Tóth, On the origin of Turing instability, J. Math. Chem. 22 (1997) 39–53.

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n-Dimensional ratio-dependent predator–prey systems with diffusion