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# Consumer resource models often have the following general form $R^{\prime}=f(R)-g(R, C) \quad C^{\prime}=\varepsilon g(R, C)-h(C)$ where R is the number of individuals of the resource and C is the number of consumers. The function $f(R)$ gives the rate of replenishment of the resource, g(R, C) describes the rate of consumption of the resource, and h(C) is the rate of loss of the consumer. The constant $\varepsilon,$ where $0<\varepsilon<1,$ is the conversion efficiency of resources into consumers. Find all equilibria of the following examples and determine their stability properties. A chemostat is an experimental consumer-resource system. If the resource is not self-reproducing, then it can be modeled by choosing $f(R)=\theta, g(R, C)=b R C,$ and $h(C)=\mu C.$ Suppose $\theta=2, b=1, \varepsilon=1, \text { and } \mu=1.$

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The goal is to find the equilibrium of a given system of differential equations and then determine whether the system of differential equations at equilibrium points is stable or unstable.

To do so, we will first replace the values:

$f\left(R\right)=2, g\left(R, C\right)=RC, h\left(C\right)=C, \varepsilon =1$

in the given system of differential equations.

We get that the system of differential equations has the form:

\begin{aligned} R'&=2-RC\\ C'&=RC-C \end{aligned}

We say that a system of two differential equations has an equilibrium at the point $(\tilde{R}, \tilde{C})$ if the differential equations satisfy:

\begin{aligned} R'&=0\\ C'&=0 \end{aligned}

at the points $R=\tilde{R}$ and $C=\tilde{C}$.

The next step is to calculate the Jacobian matrix. The Jacobian matrix of the differential system of equations has the form:

\begin{align} J(R, C)&=\begin{bmatrix} \frac{\partial f_{1}\left( R, C\right) }{\partial R}& \frac{\partial f_{1}\left( R, C\right) }{\partial C}\\[5pt] \frac{\partial f_{2}\left( R, C\right) }{\partial R}& \frac{\partial f_{2}\left( R, C\right) }{\partial C} \end{bmatrix} \end{align}

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