The following predator-prey system is discussed by May (1974): $$ \begin{aligned} &\frac{d H}{d t}=r H\left(1-\frac{H}{K}\right)-\frac{k P H}{H+D} \\ &\frac{d P}{d t}=s P\left(1-\frac{P}{\gamma H}\right) \quad \text { (parasite). } \end{aligned} $$ (host), (a) Interpret the terms appearing in these equations and suggest what the various parameters might represent. (b) Sketch the \(H\) and \(P\) nullelines on an \(H P\) phase plane. (c) Apply Bendixson's criterion and Dulac's criterion (for \(B=1 / H P\) ).

Logistic growth describes how populations change over time when limited by resources such as food or space. In the equation \( \frac{dH}{dt} = rH \left(1 - \frac{H}{K} \right) \), \( r \) is the intrinsic growth rate of the host population. The term \( 1 - \frac{H}{K} \) represents the effect of carrying capacity \( K \), where \( K \) is the maximum population size that the environment can sustain. Essentially, logistic growth shows that population growth slows as the population approaches its carrying capacity. The form \( rH \left(1 - \frac{H}{K} \right) \) ensures that growth is fast when the population is small and slows down as it approaches \( K \).

Phase plane analysis is a graphical tool used to study dynamical systems by plotting two variables against each other. For the predator-prey model, these variables are the host population \( H \) and the predator population \( P \). The nullclines are curves where the growth rate of either the host or the predator is zero. They are found by setting \( \frac{dH}{dt} = 0 \) and \( \frac{dP}{dt} = 0 \).

For the host equation, setting \( \frac{dH}{dt} = 0 \) gives:\[ rH \left(1 - \frac{H}{K} \right) - \frac{kPH}{H + D} = 0 \]. Solving for \( P \), we get a nullcline that describes where the host growth rate is zero. Similarly for the predator equation, setting \( \frac{dP}{dt} = 0 \) gives:\[ sP \left(1 - \frac{P}{\gamma H} \right) = 0 \]. Solving for \( P \), we obtain another nullcline where the predator growth rate is zero. By plotting these nullclines in the \( HP \) phase plane, we can see the interaction between the host and predator populations over time.

Bendixson's criterion is a method used to determine whether a system of differential equations has periodic solutions, such as cycles. The criterion states that if there exists a continuously differentiable function \( D(x, y) \) such that the divergence \( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \) has a constant sign (either always positive or always negative), then there are no periodic orbits within the studied region.

To apply Bendixson's criterion to our predator-prey model, calculate the divergence of the vector field formed by the system:\[ \text{div}(F) = \frac{\partial}{\partial H} \left( rH \left(1 - \frac{H}{K} \right) - \frac{kPH}{H + D} \right) + \frac{\partial}{\partial P} \left( sP \left(1 - \frac{P}{\gamma H} \right) \right) \]. If this expression has a constant sign, then Bendixson's criterion confirms that there are no periodic orbits in the phase plane.

Dulac's criterion is similar to Bendixson's criterion but uses a weight function \( B \) (often called a Dulac function) to modify the original system. It helps determine the non-existence of closed orbits. The criterion states that if there exists a continuously differentiable function \( B(x, y) \) such that the modified divergence expression \( \frac{\partial (B P)}{\partial x} + \frac{\partial (B Q)}{\partial y} \) does not change sign in a region, then there are no periodic orbits in that region.

For Dulac's criterion in our exercise, we choose \( B = \frac{1}{HP} \). Thus, we compute the modified divergence:\[ \frac{\partial}{\partial H} \left( \frac{1}{HP} \cdot P \right) + \frac{\partial}{\partial P} \left( \frac{1}{HP} \cdot Q \right) \]. If this expression consistently does not change sign, Dulac’s criterion indicates there are no periodic orbits in the predator-prey system.