Assume the discrete-time population model Nt+1 = bNt , t = 0, 1, 2 . . . Assume also

Assume a discrete-time population whose size at generation t + 1 is related to the size of the population at generation t by Nt+1 = (1.03)Nt , t = 0, 1, 2, . . . (a) If N0 = 10, how large will the population be at generation t = 5? (b) How many generations will it take for the population size to reach double the size at generation 0?

Suppose a discrete-time population evolves according to Nt+1 = (0.9)Nt , t = 0, 1, 2 . . . (a) If N0 = 50, how large will the population be at generation t = 6? (b) After how many generations will the size of the population be one-quarter of its original size? (c) What will happen to the population in the long runthat is, as t ?

Assume the discrete-time population model Nt+1 = bNt , t = 0, 1, 2 . . . Assume also that the population increases by2%each generation. (a) Determine b. (b) Find the size of the population at generation 10 when N0 = 20. (c) After how many generations will the population size have doubled?

Assume the discrete-time population model Nt+1 = bNt , t = 0, 1, 2, . . . Assume also that the population decreases by 3% each generation. (a) Determine b. (b) Find the size of the population at generation 10 when N0 = 50. (c) How long will it take until the population is one-half its original size?

Assume the discrete-time population model Nt+1 = bNt , t = 0, 1, 2, . . . Assume that the population increases by x% each generation. (a) Determine b. (b) After how many generations will the population size have doubled? Compute the doubling time for x = 0.1, 0.5, 1, 2, 5, and 10.

(a) Find all equilibria of Nt+1 = 1.3Nt , t = 0, 1, 2, . . . (b) Use cobwebbing to determine the stability of the equilibria you found in (a).

(a) Find all equilibria of Nt+1 = 0.9Nt , t = 0, 1, 2, . . . (b) Use cobwebbing to determine the stability of the equilibria you found in (a).

(a) Find all equilibria of Nt+1 = Nt , t = 0, 1, 2, . . . (b) How will the population size Nt change over time, starting at time 0 with N0?

Use the stability criterion to characterize the stability of the equilibria of xt+1 = 2 3 2 3 x2 t , t = 0, 1, 2, . . .

Use the stability criterion to characterize the stability of the equilibria of xt+1 = 3 5 x2 t 2 5 , t = 0, 1, 2, . . .

Use the stability criterion to characterize the stability of the equilibria of xt+1 = xt 0.5 + xt , t = 0, 1, 2, . . .

Use the stability criterion to characterize the stability of the equilibria of xt+1 = xt 0.3 + xt , t = 0, 1, 2, . . .

(a) Use the stability criterion to characterize the stability of the equilibria of xt+1 = 5x2 t 4 + x2 t , t = 0, 1, 2, . . . (b) Use cobwebbing to decide to which value xt converges as t if (i) x0 = 0.5 and (ii) x0 = 2.

(a) Use the stability criterion to characterize the stability of the equilibria of xt+1 = 10x2 t 9 + x2 t , t = 0, 1, 2, . . . (b) Use cobwebbing to decide to which value xt converges as t if (i) x0 = 0.5 and (ii) x0 = 3.

Rickers curve is given by R(P) = Pe P for P 0, where P denotes the size of the parental stock and R(P) the number of recruits. The parameters and are positive constants. (a) Show that R(0) = 0 and R(P) > 0 for P > 0. (b) Find lim P R(P) (c) For what size of the parental stock is the number of recruits maximal? (d) Does R(P) have inflection points? If so, find them. (e) Sketch the graph of f (x) when = 2 and = 1/2.

Suppose that the size of a fish population at generation t is given by Nt+1 = 1.5Nt e0.001Nt for t = 0, 1, 2, . . . . (a) Assume that N0 = 100. Find the size of the fish population at generation t for t = 1, 2, . . . , 20. (b) Assume that N0 = 800. Find the size of the fish population at generation t for t = 1, 2, . . . , 20. (c) Determine all fixed points. On the basis of your computations in (a) and (b), make a guess as to what will happen to the population in the long run, starting from (i) N0 = 100 and (ii) N0 = 800. (d) Use the cobwebbing method to illustrate your answer in (a). (e) Explain why the dynamical system converges to the nontrivial fixed point.

Suppose that the size of a fish population at generation t is given by Nt+1 = 10Nt e0.01Nt for t = 0, 1, 2, . . . . (a) Assume that N0 = 100. Find the size of the fish population at generation t for t = 1, 2, . . . , 20. (b) Show that if N0 = 100 ln 10, then Nt = 100 ln 10 for t = 1, 2, 3, . . . ; that is, show that N = 100 ln 10 is a nontrivial fixed point, or equilibrium. How would you find N? Are there any other equilibria? (c) On the basis of your computations in (a), make a prediction about the long-term behavior of the fish population when N0 = 100. How does your answer compare with that in (b)? (d) Use the cobwebbing method to illustrate your answer in (c). Nt+1 = Nt _ 1 + R _ 1 Nt 100 __ t = 0, 1, 2, . . .

(a) Find all equilibria when R = 0.5. (b) Investigate the system when N0 = 10 and describe what you see.

(a) Find all equilibria when R = 1.5. (b) Investigate the system when N0 = 10 and describe what you see.

(a) Find all equilibria when R = 2.5. (b) Investigate the system when N0 = 10 and describe what you see. xt+1 = rxt (1 xt ) t = 0, 1, 2, . . .

Show that for r > 1, there are two fixed points. For which values of r is the nonzero fixed point locally stable?

Use a calculator or a spreadsheet to simulate the canonical discrete-time logistic growth model with x0 = 0.1 for t = 0, 1, 2, . . . , 100, and describe the behavior when (a) r = 3.20 (b) r = 3.52 (c) r = 3.80 (d) r = 3.83 (e) r = 3.828 Nt+1 = R(Nt )Nt R(N) R(N)N N > 0r > 0K > 0 > 1

In Problems 2325, we consider density-dependent population growth models of the form Nt+1 = R(Nt )Nt The function R(N) describes the per capita growth. Various forms have been considered. For each function R(N), find all nontrivial fixed points N (i.e., N > 0) and determine the stability as a function of the parameter values. We assume that the function parameters are r > 0, K > 0, and > 1. R(N) = rN1

In Problems 2325, we consider density-dependent population growth models of the form Nt+1 = R(Nt )Nt The function R(N) describes the per capita growth. Various forms have been considered. For each function R(N), find all nontrivial fixed points N (i.e., N > 0) and determine the stability as a function of the parameter values. We assume that the function parameters are r > 0, K > 0, and > 1. R(N) = r 1 + N/K

In Problems 2325, we consider density-dependent population growth models of the form Nt+1 = R(Nt )Nt The function R(N) describes the per capita growth. Various forms have been considered. For each function R(N), find all nontrivial fixed points N (i.e., N > 0) and determine the stability as a function of the parameter values. We assume that the function parameters are r > 0, K > 0, and > 1. R(N) = er (1N/K)