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MATH3506 Mathematical Ecology Problem Sheet 2

1. Find N(≠), the solution to the time-dependent equation

 = A cos(w≠) N ╱ 1 一 ← ,   A, K > 0 and w are constants.

What can you say about N(≠) as ≠ → &?  What are the maximum and minimum populations?

2. A population with continuous generations has constant per capita death-rate d(N) = 6 > 0, but density dependent per capita birth-rate, ó = 8(N), where

γN   

8(N) =

Plot the function 8 as a function of N > 0. Observe that the associated net-reproductive rate o(N) does not satisfy the assumption o\ (N) < 0.  Why might you nevertheless regard these conditions as biologically reasonable?  [Think about how very small popu- lations might be distributed in their territory, and what consequences this might have for their breeding success.]

dN

Study the dynamics of this population, i.e.  according to         = N(8(N) 一 6).  Find

the possible steady states and their stability for different values of γ, 6, K, and make an ecological interpretation of your results.

3. The population of deers in a forest follows a logistic growth with positive constant intrinsic growth rate γ and constant carrying capacity K . A new settlement of humans in the forest is starting to hunt a constant number of deers per day, enough to feed the settlement. This reduces the total population growth of the deers with a rate 9 .

(a) Write an equation for the evolution in time of the population of deers in the forest dN/d≠ .

(b) Plot schematically the population growth rate as a function of the population size for different values of 9 .  Which different behaviours are expected for different values of 9?

(c) What is the maximum hunting rate 9h  such that the deer population does not collapse?