In May (1978), Host-parasitoid systems in patchy environments: a phenomenological model. J. Anim. Ecol., \(47,833-843\), the following system of equations appear: $$ \begin{aligned} &H_{r+1}=F H_{1}\left(1+a P_{1} / k\right)^{-k}, \\ &P_{r+1}=H_{1}-H_{r+1} / F . \end{aligned} $$ Determine the steady states and their stability for this system. (The parameters \(F, a, k\) are positive.)

In the context of the host-parasitoid model, steady state analysis means finding points where populations of hosts (H) and parasitoids (P) do not change over time. These points are known as steady states or equilibrium points.
To perform a steady state analysis, you set the equations for both host and parasitoid populations equal to their values in the next generation.
This ensures that there is no change in the population size from one time step to the next.

Formally, for the given equations:
\[ H_{r+1} = F H_{r}(1 + a P_r / k)^{-k} \] and \[ P_{r+1} = H_r - H_{r+1} / F \]

We set \( H_{r+1} = H_r \) and \( P_{r+1} = P_r \).
This allows us to solve for population levels where the dynamics are in balance.

A system of equations involves multiple equations that must be solved together.
In this context, the host-parasitoid model consists of two equations representing two populations.
These equations need to be solved simultaneously to find the relationship between hosts and parasitoids over time.

The given system is:
\[ H_{r+1} = F H_{r}(1 + a P_r / k)^{-k} \] and \[ P_{r+1} = H_r - H_{r+1} / F \]

Understanding and simplifying these equations helps to find the steady states.
In practice, you would start by isolating variables and making substitutions to reduce the complexity of the system.
This will help you solve for specific population levels (values for \(H\) and \(P\)) that satisfy both equations.

Once steady states are found, the next step is to determine their stability.
Stability analysis checks if a small perturbation or change in population will return to the steady state or diverge away from it.

If a steady state is stable, small changes will diminish over time and the system will return to equilibrium.
If unstable, small changes will grow over time, causing the system to move away from equilibrium.

The stability of each steady state can be determined by examining the Jacobian matrix (a tool that helps analyze stability).
This involves calculating partial derivatives of the system of equations with respect to each population variable and evaluating them at the steady states.

The Jacobian matrix is essential for stability analysis.
It consists of partial derivatives that help understand how changes in one variable affect others in a system.

For a system with host and parasitoid populations, the Jacobian matrix J is:
\[ J = \begin{pmatrix} \frac{\text{d}H_{r+1}}{\text{d}H_r} & \frac{\text{d}H_{r+1}}{\text{d}P_r} \ \frac{\text{d}P_{r+1}}{\text{d}H_r} & \frac{\text{d}P_{r+1}}{\text{d}P_r} \end{pmatrix} \]

Each element of this matrix represents how one population affects the next generation of both hosts and parasitoids.

Evaluating this matrix at the steady states gives information on stability.
The values in the matrix help to find eigenvalues, which provide further insights.

Eigenvalues are numerical values derived from the Jacobian matrix that help determine stability.
They give insight into how the system's equilibrium reacts to small changes.

For a steady state to be stable, all eigenvalues must have negative real parts.
If any eigenvalue has a positive real part, the steady state is unstable.
Zero real parts typically indicate a special case or require further analysis.

To find the eigenvalues, compute the characteristic equation of the Jacobian matrix and solve for its roots.
This process helps categorize the behavior near equilibrium points and informs decisions about stability in biological systems.