Mathematics, 08.04.2020 01:36 GamerGirl15

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.

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Answer from: epicriderexe

Thus, the proportion of time is restricted to:

$\pi = 0.208$

Given that an exponential rate $\lambda$ , at an expoential rate $\mu$ . An exponential rate of increase $\theta$ : the popuilation size is N.

Hence the total birth rate where there are n persons in the system is: $n\lambda+\theta$ . are assume to accury at an expoential rate $\mu$ for each member of the population. So: $n\mu = \mu n$

Let x(t) denote the population size at time t. Suppose: x(0)= ix(0)= i and let M(t)= E[x(t)] . 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:

M(t+h)= E[x(t+h)]= E(E[x(t+h)] x(t)) .

Now, given the size of the population at the time then, ignoring events whose probability is $\theta (h)$ . 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 x(t)*x(t) :

x(t+h)x(t+h)=x(t)+1 with pobability

$[\theta +x(t)x]h +\theta h \\$

X(t)-1 with probability $x(t)\mu h +\theta (h)$

x(t) with probability $1-[\theta + x(t) \lambda + x(t) \mu ] h + \theta(h)$

$E[x(t+h)x(t)]= x(t)+[\theta+ x(t)\lambda-x(t)\mu]h+\theta(h)$

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

For $0\leq i\leq N$

we have $\lambda i = i \lambda + \theta$

For $i \geq N$ we have $\mu i = i \mu$

$\pi$ has $\pi \theta = \frac{1}{1+ \sum \phi n }$

where $\phi n = \frac{\lambda}{\mu}$

When $\theta=\lambda$

For $n\geq 3$ we see $\phi n + \frac{1}{\phi n } = \frac{n \lambda}{(n+\mu) \mu}$

This leads to $\pi = 0.208$

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Each individual in a biological population is assumed to give birth at an exponential rate λ and to

Answer from: oliviavaughan01

π0 = 0.208

Step-by-step explanation:

check the attachment for explicit explanations.

Answer from: Quest

step-by-step explanation:

you are throwing one dart

Answer from: Quest

this is a very good question

step-by-step explanation:

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