Question

POPULATION MODELS: PLEASE ANSWSER ASAP: ALL 3 AND WILL RATE U ASAP.

The logistic growth model describes population growth when resources are constrained. It is an extension to the exponential growth model that includes an additional term introducing the carrying capacity of the habitat The differential equation for this model is P(t) Where P(t) is the population (or population density) at time t, k > 0 is a growth constant, and M is the carrying capacity of the habitat. This model has two steady states: one when P-0 and one when P-M 1. [1 Is this differential equation time-invariant? Why/why not?l 2. [4] Let's modify the logistic model to include a new term for hunting/predation. Consider the differential equation P(t)c where P, M, k are defined as before and c>0 is a predation term removing some of the population over each time unit Find the steady-state solution(s) (i.e0) to this differential equation. For each solution, provide a brief interpretation (in terms of animals, carrying capacity, etc.) of how the various factors contribute to a zero net population change. 3. [2] Suppose you are managing a shelter for a vulnerable species whose population dynamics can be described by the DE in Question 2 with t representing a time in years Each year you are planning to gradually release c- 200 animals from your shelter back into the wild. Assuming your shelter has a carrying capacity of M1000 animals with growth constant k = 1, how many animals would this shelter need initially in order to maintain a steady population? If there are multiple solutions, briefly describe the merits of each one.

The logistic growth model describes population growth when resources are constrained. It

is an extension to the exponential growth model that includes an additional term introducing

the carrying capacity of the habitat.

The differential equation for this model is:

dP/dt=kP(t)(1-P(t)/M)

Where P(t) is the population (or population density) at time t, k > 0 is a growth constant,

and M is the carrying capacity of the habitat. This model has two steady states: one when

P = 0 and one when P = M.

1. [1] Is this differential equation time-invariant? Why/why not?

2. [4] Let's modify the logistic model to include a new term for hunting/predation.

Consider the differential equation

dp/dt=kP(t)(1-P(t)/M)-c

where P;M; k are defined as before and c > 0 is a predation term removing some of

the population over each time unit.

Find the steady-state solution(s) (i.e. dP/dt = 0) to this differential equation. For each

solution, provide a brief interpretation (in terms of animals, carrying capacity, etc.) of

how the various factors contribute to a zero net population change.

3. [2] Suppose you are managing a shelter for a vulnerable species whose population

dynamics can be described by the DE in Question 2 with t representing a time in years.

Each year you are planning to gradually release c = 200 animals from your shelter back

into the wild. Assuming your shelter has a carrying capacity of M = 1000 animals

with growth constant k = 1, how many animals would this shelter need initially in

order to maintain a steady population? If there are multiple solutions, briefly describe

the merits of each one.

Answer 1

• 
  Steady state solution = dP/dt = 0 = > KP(t) [1 - P(t)/M] - C = 0 dP/dt = 0 = > �P� is invariant with time = > P(t) = P KP [1 - P/M] - C = 0 KMP - KP^2 - MC = 0 KP^2 - KMP + MC = 0 P_1P_2 = KM PlusMinus squareroot K^2M^2 - 4KMC/2K dP/dt = -[P-P_1][P-P-2] Suppose P_1 = KM + squareroot K^2M^2 - 4KMC/2K dP/dt > 0 when P_2 < P < P_1 dP/dt < 0 when P > P_1 or P < P_2 Population is increasing with time when population is in between P_1 and P_2 Population is decreasing with time when population is either greater than P_1 and less than P_2 The steady state solutions p_1, P_2 are invariant with time but they depend upon the values of growth constant K and carrying capacity of the habitat M To maintain steady population dP/dt = 0 solutions for this are P_1
  

The differential equation given is not invariant with time.