Chapter 7.6, Problem 2E

(a)

To determine

To Find: Whether the system describes the completion of cooperation and the explanation for why it is reasonable model.

Answer to Problem 2E

The $c_1=0.00001>0$ and $c_2=0.00004>0$ shows that the relation is cooperative.

Explanation of Solution

Given:

The given equation are,

  $\begin{array}{l}\frac{dx}{dt}=0.12x-0.0006x^2+0.00001xy\\ \frac{dy}{dt}=0.08x+0.00004xy\end{array}$

Calculation:

Consider the given system is,

  $\begin{array}{l}\frac{dx}{dt}=a_{1}x-b_1{x}^2+c_{1}xy\\ \frac{dy}{dt}=a_{2}x-b_2{y}^2+c_{2}xy\end{array}$

The system is said to be cooperative if $c_1>0$ and $c_2>0$ . The equation is cooperative for $c_1>0$ and $c_2>0$ , it is competitive if $c_1<0$ and $c_2<0$ . It is predator if-prey if $c_1{c}_2<0$ .

Consider the given equation is,

  $\begin{array}{l}\frac{dx}{dt}=0.12x-0.0006x^2+0.00001xy\\ \frac{dy}{dt}=0.08x+0.00004xy\end{array}$

Compare the given expression with the standard form as,

  $\begin{array}{l}{c}_1=0.00001>0\\ c_2=0.00004>0\end{array}$

Consider that the $c_1,c_2$ is positive this shows that the other species is beneficial for both, this shows that the relationship is cooperative.

(b)

To determine

To Find: Whether the system describes the completion of cooperation and the explanation for why it is reasonable model.

Answer to Problem 2E

The $c_1=-0.0006<0$ and $c_2=0.0002<0$ shows that the relation is completive.

Explanation of Solution

Given:

The given equation are,

  $\begin{array}{l}\frac{dx}{dt}=0.15x-0.0002x^2-0.00006xy\\ \frac{dy}{dt}=0.00008y^2-0.0002xy\end{array}$

Calculation:

Consider the given system is,

  $\begin{array}{l}\frac{dx}{dt}=a_{1}x-b_1{x}^2+c_{1}xy\\ \frac{dy}{dt}=a_{2}x-b_2{y}^2+c_{2}xy\end{array}$

The system is said to be cooperative if $c_1>0$ and $c_2>0$ . The equation is cooperative for $c_1>0$ and $c_2>0$ , it is competitive if $c_1<0$ and $c_2<0$ . It is predator if-prey if $c_1{c}_2<0$ .

Consider the given equation is,

  $\begin{array}{l}\frac{dx}{dt}=0.15x-0.0002x^2-0.00006xy\\ \frac{dy}{dt}=0.00008y^2-0.0002xy\end{array}$

Compare the given expression with the standard form as,

  $\begin{array}{l}{c}_1=-0.0006<0\\ c_2=0.0002<0\end{array}$

Consider that the $c_1,c_2$ is negative this shows that the other species is beneficial for both, this shows that the competitive is cooperative.