Each individual in a biological population is assumed to give birth at an exponential rate λ and to die at an exponential rate µ. In addition, there is an exponential rate of increase θ due to immigration. However, immigration is not allowed when the population size is N or larger. a) Set this up as a birth and death model. b) If N=3, λ=θ=1, and μ=2, determine the proportion of time that immigration is restricted.

Thus, the proportion of  time is restricted to:

Given that an exponential rate ,  at an expoential rate . An exponential rate of increase : the popuilation size is N.

Hence the total birth rate where there are n persons in the system is: .  are assume to accury at an expoential rate for each member of the population. So:

Let x(t) denote the population size at time t. Suppose: and let . So they will determine M(t) by deriving and then solving a differential equation that is :

We start by deriving on equation for M(t+h) by conditioning on X(t) this yelds:

.

Now, given the size of the population at the time then, ignoring events whose probability is . The population at time t+h. Will be either increase in size by 1. If a birth or immigration occurs in (t, t+h) or decrease by 1 if a death occurs in this internal, or remain the same if neither of these two possibilities occurs that is given :

 with pobability

X(t)-1 with probability

x(t) with probability

Since the probabilities have to add up to 1, that determines. The value of the probabiliy measure for he best case.

For

we have

For we have

has

where

When

For we see

This leads to

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π0 = 0.208

Step-by-step explanation:

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