Problem 1: Harvesting prevents the population size of a species from attaining its natural carrying capacity. We can add harvesting to the logistic model by assuming that the per capita harvest rate is m per day in a population whose intrinsic growth rate is r per day and whose carrying capacity is K in the absence of harvesting. (a) Derive a differential equation describing the dynamics of the population size. (b) Determine the equilibria for this modeL (c) Determine the stability of these equilibria. (d) What condition must hold for the population to persist? (e) What is the maximum allowable harvest rate that ensures that the population size will remain stable at a size greater than 1000, which is considered by some to represent a minimum viable population size?
Problem 2: Population size might be regulated by competition for suitable territories. Consider a large number of suitable territories or patches. At time t, a fraction p(i) of these patches are occupied. Of the unoccupied sites, a fraction mp(t) are recolonized from occupied patches. Subsequently, each occupied site suffers a risk of local extinction e through catastrophic events such as fire or disease. These assumptions are consistent with the following discrete-time recursion equation for the fraction of occupied sites:
p(t + 1) = (1-e)(p(t) + mp(t)(1-p(t))).
(a) Find the two equilibria of this modeL (b) Under what conditions is there a biologically valid equilibrium with the species present? (c) Given that the equilibrium in (b) is valid, when is it stable? (d) If we assume that in is less than one so that not all patches can be immediately recolonized, can the fraction of occupied sites overshoot the equilibrium?
Problem 3: The extinction-recolonization model described in Problem 5.12 can also be studied in continuous time, using the differential equation
dp/dt = mp(1 - p) - e p.
(a) Find the two equilibria of this modeL (b) Under what conditions is there a biologically valid equilibrium with the species present? (c) Given that the equilibrium in (b) is valid, when is it stable? (d) Is it possible for the fraction of occupied sites to overshoot the equilibrium?