# Differential Equations : Predator / Prey Models

Part a)

Given the following predator prey model where x(t) is the predator population and y(t) is the prey population:

dx/dt = - ax + bxy + (z1)*x

dy/dt = cy - gxy +(z2)*y

Here both z1 and z2 can be positive or negative; parameters a, b, c, g are all defined to be positive.

Parameters z1 and z2 can represent an external introduction of species at a constant rate (positive values) or an external harvesting of the species at a constant rate (a negative values).

Under what conditions of z1 and z2 can the trivial fixed point (both species evolve to extinction) become stable (in both directions not a saddle node)

Interpret these results.

Part b)

Given the following predator prey model where x(t) is the predator population and y(t) is the prey population:

dx/dt = - a*x + b*xy -----Q

dy/dt = c*y - g*xy ------P

Solve the second equation, equation P, for the predator population (x) and simplify. Then show that the average predator population over one cycle or period T is equal to c/g.

Note that the populations are periodic so y(T) = y(0) and x(T) = x(0)

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#### Solution Summary

Differential equations and parameters are used to analyze a predator / prey model. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

Population Dynamics

I think the solution is probably a simple modification to the predation equations:

dN/dt = rN - cNP is the growth rate for the prey population and

dP/dt = -dp + gcNP is the growth rate for predator population, where:

N= prey density

P = predator density

d= death rate

g = conversion efficiency of prey to predators

r = per capita growth rate of prey

c = rate that predator eats

My latest idea is to combine these equations to get:

dP1/dt = -dP1 + cP2P1+rP2-cP21P1 and

dP2/dt = -dP2 +cP1P2 +rP1-cP1P2

does that sound reasonable to any of you math experts out there?

I need to use the basic Lotka-Volterra equations below to represent a situation where there are 2 predators that eat each other (rather than a single predator eating a single prey). And using those equations determine what happens to the growth rate of each predator over time. Does that help?

I am trying to figure out how to write a pair of differential equations for 2 predators that eat each other. This should assume that there is no density dependence, each has a separate and constant death rate, each increases solely as a result of consumption of the other, and each has a type 1 functional response to the other.

I do not know where to start. We have covered the Lotka -Volterra equations which represent either competition or predation, which I understand, but I am just not sure how to represent the above situation mathematically.

So far my ideas are:

1) combining the equations for competition and predation, since both

are now predators and they will be competing with one another:

So a type 1 functional response would be represented by:

dN/dt = rN - cNP is the growth rate for the prey population and

dP/dt = -dp + gcNP is the growth rate for predator population, where:

N= prey density

P = predator density

d= death rate

g = conversion efficiency of prey to predators

r = per capita growth rate of prey

c = rate that predator eats

and the competition eqn's are:

dN1/dt = r1N1 * (1- N1/K1 - a*N2/K1) and

dN2/dt = r2N2 * (1- N2/K2 - b*N1/K2).

where K is carrying capacity and

a & b are competition coefficient

s. I don't think I can simply combine these since a type 1 response shows exponential growth while the competition equations take density dependence into account. Somehow it seems I should be able to treat both as predator AND prey. So my next idea is to create an equation that would represent a mutualistic situation (where both benefit) or perhaps commensialism (where one benefits and the other neither harms nor benefits. However, I am not sure)

Once I come up with the equation I also need to make a discrete time step version of this model to see if it behaves similarly. I assume I would start by N(t=1) =

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