Chapter 7, Problem 8RCC

a.

To determine

To write : the lokta-volterra equation

Answer to Problem 8RCC

The equation for fish

The equation for shark

Explanation of Solution

Given information : the question is

To give the equation for model population of food fish and sharks

Here fish $\left(F\right)$ is the prey and the Shark $\left(S\right)$ is the predator

The lotka-volterra equations to model their population as

  $\begin{array}{l}\frac{dF}{dt}=kF-aFS\\ \frac{dS}{dt}=-rS+bFS\end{array}$

Where $a,b,r,k$ are the positive constants

b.

To determine

What do these equations say about each population in the absence of other

Answer to Problem 8RCC

The population of fishes increases in the absence of sharks

The population of sharks decreases in the absence of fishes

Explanation of Solution

Given information : The question is

To give the equation in which the population of fishes and sharks absence

from part (a)

  $\begin{array}{l}\frac{dF}{dt}=kF-aFS\mathrm{......}\left(1\right)\\ \frac{dS}{dt}=-rS+bFS\mathrm{......}\left(2\right)\end{array}$

In the absence of sharks

  $\begin{array}{l}S=0\\ \frac{dF}{dt}=kF\end{array}$

Where $k$ is a positive constant

So, the population of fishes increases in the absence of sharks

In the absence of fishes

  $\begin{array}{l}F=0\\ \frac{dS}{dt}=-rS\end{array}$

Where $r$ is a positive constant

So, the population of sharks decreases in the absence of fishes