We have seen a predator-prey system in which a lack of predators results in the

Question

We have seen a predator-prey system in which a lack of predators results in the prey population growing according to an exponential population model. You will model more complex predator-prey systems as described below. For each system, and any equilibrium solutions, draw a vector field and a direction field, and describe qualitatively the behavior of the solutions.

Explain in words why your descriptions are consistent with the assumptions

of the system. While you are free in each system to choose your own (reasonable)

values for the parameters (, , , , N, M, k), you should discuss the ways in which changing those values would significantly alter the system, if any. Your work should be cohesive and easily read.

The systems you should model are:

1. A predator-prey system as discussed in class, except that a lack of predators results in the prey population growing according to a logistic model with carrying capacity N.

2. A predator-prey system as discussed in class, except that a lack of predators results in the prey population growing according to a modified logistic

model with carrying capacity N and sparsity constant M.

3. A predator-prey system as discussed in class, with the additional assumption that prey moves away from the area at a rate proportional to the size of the predator population with proportionality constant k.

Explanation / Answer

A predator-prey system except that a lack of predators results in the prey population growing according to a logistic model with carrying capacity N.

Population Growth Models  

DN/Dt=  B – D + I – E  

N = population size

t = time  

B = # of births  

D = # of death  

I = # of immigrants  

E = # of emigrants  

Growth rate per capita:  

DN/Dt= BN- DN

(b – d) N = r N  

B = birth rate  

D = death rate  

R = the intrinsic rate of natural increase in population size

Therefore the Final equation the predator-prey system  

growth rate at a particular time (G): dN/dt = r N

A predator-prey system except that a lack of predators results in the prey population growing according to a modified logistic model with carrying capacity N and sparsity constant M.

No population can grow exponentially forever.

Carrying capacity (K):

Maximum population size that a particular environment can support As M increases, (K-M) K approaches 0.

A predator-prey system with the additional assumption that prey moves away from the area at a rate proportional to the size of the predator population with proportionality constant k will be

(K–M) K = fraction of K that is still available for growth (that the environment can still accommodate) L.

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