Discrete Systems - fish population modelling question on derivation of the model

Question

I am currently reading 'Introduction to Non-Linear Systems' by J Berry and on page 112 there is an exercise (question 5) which I don't understand, it is:

The population of fish in a large lake has been stable for some time. Prior to this situation the population was decreasing from an initially relatively high level. When the population was 4000 the proportionate birth rate was 10% and the proportionate death rate was 70%. When the population was 3000 the proportionate birth rate was 30% and the proportionate death rate was 60%. This population model also has the following assumptions:

• there is no exploitation and no restocking
• the proportionate birth rate is a decreasing linear function of the population
• the proportionate death rate is an increasing linear function of population

Show that the model based on these assumptions and the above data predicts that the populations falls according to the logistic model and find the equilibrium population size

Now the book has given the following solution:

$$P_{n+1}-P_n = \left (0.6 - 3 \times 10^{-4}P_n \right)P_n$$ and that the stable equilibrium is 2000 fish.

Now my question is how is this model derived from the data in the question? And how is the equilibrium calculated?

I think perhaps the solution above is using the conservation law which is $$accumulation = input - output$$ But I can't get from the numbers in the question to a formula like that above.

Answer 1

This has the form $P_{n+1}-P_n=r(P_n)P_n$ where $r(P)$, positive or negative, is the reproduction rate at population $P$. One is given that $r(4000)=10\%-70\%$ and $r(3000)=30\%-60\%$. The linearity assumptions imply that $r(P)=b-aP$ for some $a$ and $b$ independent of $P$. Solving the system of two equations $b-4000a=-0.6$ and $b-3000a=-0.3$ yields $a=0.0003$ and $b=0.6$, as desired.

The points $P_*$ of equilibrium are when $P_n=P_*$ yields $P_{n+1}=P_*$. Here, the equilibrium you are asked to detect is when $r(P_*)=0$, that is, at $P_*=b/a=2000$, but note that there is a second one at $P=0$. The next question in the exercise might be to show that $P_*=2000$ is attractive while $0$ is repulsive.