It is known that if the deer population falls below a certain level, m, then the deer will become extinct. It is also known that is the deer population goes above the maximum carrying capacity, M, the population will decrease to M.
(a) Discuss the reasonableness of the following model for the growth rate of the deer population as a function of time: dP/dt = kP(M-P)(P-m), where P is the population and k is a constant of proportionality.
(b) Explain how this growth rate model differs from the logistical model dP/dt = kp(M-P) Is it better or worse than the logistical model?
(c) Show that if P > M for all t, then the limit as P(t) approached infinity is M.
(d) Assuming m < P < M for all t, briefly explain the steps you would use to solve the differential equation.
(e) What are the equilibrium point of the model? Explain the dependence of the equilibrium level of P on the initial conditions. How many deer hunting permits should be issued?
(a) The given model for the growth rate of the deer population, dP/dt = kP(M-P)(P-m), is reasonable for several reasons. Firstly, it takes into account the carrying capacity of the environment by incorporating the term (M-P), which represents the available resources or space. Secondly, the term (P-m) represents the minimum population level required to avoid extinction. This reflects the fact that if the population falls below this level, the deer will become extinct. Lastly, the model includes the constant of proportionality k, which determines the rate at which the population grows or declines.
(b) The growth rate model, dP/dt = kP(M-P)(P-m), differs from the logistic model, dP/dt = kp(M-P), in the sense that it incorporates an additional factor, (P-m), which represents the minimum population level required to avoid extinction. This inclusion makes the growth rate model more sophisticated as it accounts for a critical threshold below which the population will decline towards extinction. However, whether this makes the growth rate model better or worse than the logistic model depends on the specific context and the desired level of accuracy in the prediction of the deer population.
(c) To show that if P > M for all t, then the limit as P(t) approaches infinity is M, we can analyze the behavior of the growth rate model. When P > M, the term (M-P) is negative, and as P increases towards infinity, the term (M-P) tends to -∞. Since (M-P) is multiplying the whole expression, the population growth rate (dP/dt) becomes increasingly negative as P increases. This causes the population to decline towards the carrying capacity M, ultimately reaching a stable equilibrium at M.
(d) Assuming m < P < M for all t, to solve the differential equation dP/dt = kP(M-P)(P-m), we would start by separating variables and integrating both sides of the equation. By doing so, we aim to find a function P(t) that satisfies the equation for a given initial condition.
(e) The equilibrium points of the model can be found by setting the growth rate, dP/dt, equal to zero and solving for P. However, to analyze the dependence of the equilibrium level of P on initial conditions, we need additional information such as the specific values of k, m, and M. Based on the equilibrium points, we can then determine the stability of the system and make recommendations on the number of deer hunting permits to be issued, considering the desired population size and the carrying capacity of the environment.