Modeling Predator Prey Systems Fin

8/13/2019 Modeling Predator Prey Systems Fin

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Pablo David GonzalezM Angeles Lezcano

Nuria ValdesPredictive Limnology (IWQA)

3/16/2010


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1. Background information

1.1 Modeling ecosystem population

1.2 Carrying capacity constraints1.3 The Lotka-Volterra model

2.Difference equations and the steady-state

solution

3. Modeling the dynamic Deer-Wolf system


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The size of a population depends on births and deaths.

Births and deaths population density n/AIf the area is fixed size of the population in this fixed area.

Number of births tends to increase as the population in a fixedarea increases a higher population density provides more ofan opportunity for mating among individuals in the population.

B(t)=bP(t) b=births per capita per unit timeP=number of individuals

Deaths increase as the population increases intraspeciescompetition

D(t)=dP(t) d=deaths per capita per unit time


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Rate at which the population changesover time is proportional to the sizeof the population.

Constant of proportionality: (b -d) difference between births anddeaths rates

=net growth rate = b - d

Exponential growth/decay functionno resource limitation constrains thenet growth rate


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N(t) populations depends on the resources in theecosystem for survival N(t) cannot exceed a certainsize carrying capacity

KIf b > d population will grow and eventuallyapproach K intraspecies competition

death rate begins to increase towards avalue equal to the birth rate

moves to zero when it reacheszero, b = d, the population levels off at the carryingcapacity


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Now, lets suppose that birth rate remains constantand death rate is a function of population andcarrying capacity:

b = r

Logistic growth


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Elaborated by Alfred J. Lotka (1925) and Vittora Volterra(1926).

Based on the predator-prey relationship.

There are some previous conditions: fixed area ecosystem so population size

depends on the population density (it means thatgrowth is density dependent)

The only predators food supply is the population ofprey that exists in the ecosystem predators aremonophagus.

There is no immigration or emigration of predator orprey species into or out of this ecosystem.

There is a carrying capacity of the ecosystem.


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ases- If there is no predator population, prey population would

behave according to the logistic behaviour patternbecause it is density dependent: as soon as preypopulation increases, net growth rate decreases as thereis intraspecies competition.

The system regulates itself toward a stable size near thecarrying capacity of the ecosystem.


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Cases

- If we add a predator population, the net growth rate ofpreys (N) decreases. It means that the higher thepredators, the less the net growth rate of preys. So,population of predators is related to the death rate ofthe prey population.

Dd= death rate of prey due to the predatory wolves.

N

P


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If we add preys to the model, the net growth rate of

predator (P) increases. It means that, the higher thenumber of preys, the higher the net growth rate ofpredators. So, population of predator depends onpopulation of preys.

Wb = inflow to the wolf population which relates prey

population and birth rate of wolves.

P

P


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Once we have added wolves to the model, we have totake into account two things:

- Efficiency of predators to catch and kill prey. c

- Efficiency of predators to translate prey kills into newpredator individuals.

So:

Wd= decline of natural wolf population


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We want to predict whether a perturbation tothe model will result in stability or instability

The rate of change ofthe stock variable is not

0 and it is a density-

dependent term

Prey population

Predator population


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In steady-satate conditions the rate ofchange of the stock variable equals zero

To simplify the model we remove the density-

depent variable infinite carrying capacity

All death


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If we increase r (more fecundity of deerpopulation) wolves population will increase

Reason: wolf population increase to consume

additional deer that are born.

If we increase Wd(death rate of wolves)

deer population will increaseReason: Because there are more deers, productionof wolves will increase taking place of those thatdie


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Lotka-Volterra phase space diagram

Prey population

Predator

population

A

B C

D

Prey isocline

Predator isocline

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Modeling Predator Prey Systems Fin