Exercises

Question 1:

All the following discrete time population models are of the form N_t+1 = f(N_t) and have been taken from the ecological literature and all have been used in modelling real situations. Determine the nonnegative steady states, discuss their linear stability and find the first bifurcation values of the parameters, which are all taken to be positive.

(i) N_t+1 = N_t[1+ r (1 - N_t/K)],

(ii) N_t+1 = rN_t^1-b,  if N_t > K,
 = rN_t, if N_t < K,

(iii) N_t+1 = rN_t/(1 + aN_t)^b

(iv) N_{t +1} = rN_t/(1 + (N_t/K))^b

Question 2 Construct cobweb maps for:

(i) N_t+1 = ((1 + r)N_t)/(1 + rN_t)

(ii) N_t+1 = rN_t/(1 + aN_t)^b, a>0, b>0, r>0

and discuss the global qualitative behaviour of the solutions. Determine, where possible, the maximum and minimum N_t, and the minimum for (ii) when b << 1.

Question 3 Verify that an exact solution exists for the logistic difference equation

u_t+1 = ru_t(1 - u_t), r > 0

in the form u_t = A sin²α^t at by determining values for r, A and α. Is the solution (i) periodic? (ii) oscillatory? Describe it! If r > 4 discuss possible solution implications.

Question 4 The population dynamics of a species is governed by the discrete model

N_t+1 = f(N_t) = N_texp [r (1 - N_t/K)]

where r and K are positive constants. Determine the steady states and their corresponding eigenvalues.

Show that a period-doubling bifurcation occurs at r = 2. Briefly describe qualitatively the dynamic behaviour of the population for r = 2 + ε, where 0 < ε << 1. In the case r > 1 sketch N_t+1 = f(N_t) and show graphically or otherwise that, for it large, the maximum population is given by  N_m = f (K/r) and the minimum possible population by N_m = f (f(K/r). Since a species becomes extinct if N_t ≤ 1 for any t > 1, show that irrespective of the size of r > 1 the species could become extinct if the carrying capacity K < r exp [1 + e^r-1 -2r].

Question 5

The population of a certain species subjected to a specific kind of predation is modelled by the difference equation

u_t+1 = a(u_t²/(b² + u_t²), a > 0

Determine the equilibria and show that if a² > 4b² it is possible for the population to be driven to extinction if it becomes less than a critical size which you should find.

Question 6 It has been suggested that a means of controlling insect numbers is to introduce and maintain a number of sterile insects in the population. One such model for the resulting population dynamics is

N_t+1 = RN_t²/((R - 1)N_t²/M + N_t + S)

where R > 1 and M > 0 are constant parameters, and S is the constant sterile insect population.

Determine the steady states and discuss their linear stability, noting whether any type of bifurcation is possible. Find the critical value Se of the sterile population in terms of R and M so that if S > S_c the insect population is eradicated. Construct a cobweb map and draw a graph of S against the steady state population density, and hence determine the possible solution behaviour if 0 < S < Sc.

Question 7 A discrete model for a population N_t, consists of

N_t+1 = rN_t/(1 + bN_t²) = f(N_t),

where t is the discrete time and r and b are positive parameters. What do r and b represent in this model? Show, with the help of a cobweb, that after a long time the population N_t is bounded by

N_min = 2r²/(4 + r²)√b ≤ N_t ≤ r/2√b

Prove that, for any r, the population will become extinct if b > 4.

Determine the steady states and their eigenvalues and hence show that r = 1 is a bifurcation value. Show that, for any finite r, oscillatory solutions for N_t are not possible.

Consider a delay version of the model given by

N_t+1 = rN_t/(1 + bN_t-1²) = f(N_t), r > 1

Investigate the linear stability about the positive steady state N* by setting N_t = N* + n_t. Show that n₁ satisfies

n_t+1 - n_t + 2(r -1)r^-1n^t-1 = 0

Hence show that r = 2 is a bifurcation value and that as r → 2 the steady state bifurcates to a periodic solution of period 6.

Question 8 A basic delay model used by the International Whaling Commission (IWC) for montioring whale populations is

u_t+1 = Su_t + R(u_t-T), 0 < s < 0,

where T ≥ 1 is an integer.

(1) If u* is a positive equilibrium show that a sufficient condition for linear stability is |R'(u*)| < 1 - s. [Hint: Use Rancho's theorem on the resulting character¬istic polynomial for small perturbations about u*.]

(ii) If R(u) = (1 - s)u[1 + q(1 - u)], q > 0 and the delay T = 1, show that the equilibrium state is stable for all 0 < q < 2. [With this model, T is the time from birth to sexual maturity, s is a survival parameter and R(u_t - T) the recruitment to the adult population from those born T years ago.]

Question 9 Consider the effect of regularly harvesting the population of a species for which the model equation is

u_t+1 = bu_t²/(1 + u_t²) - Eu_t = f(u_t; E), b > 2, E > 0,

where E is a measure of the effort expended in obtaining the harvest, E_ut,. [This model with E = 0 is a special case of that in Exercise 5.]

Determine the steady states and hence show that if the effort E > E_m = (b - 2)/2, no harvest is obtained. If E < E_m show, by cobwebbing u_t+1 = f (u_t; E) or otherwise, that the model is realistic only if the population u_t always lies between two positive values which you should determine analytically.

With E < E_m evaluate the eigenvalue of the largest positive steady state. Demonstrate that a tangent bifurcation exists as E → E_m.