Problem 40 - Calculus for Biology and Medicine

Question

Find all biologically relevant equilibria of the negative binomial host- parasitoid model $$ \begin{array}{l} N_{t+1}=4 N_{t}\left(1+\frac{0.01 P_{t}}{2}\right)^{-2} \\ P_{t+1}=N_{t}\left[1-\left(1+\frac{0.01 P_{t}}{2}\right)^{-2}\right] \end{array} $$ and analvze their stability.

Step-by-Step Solution

Answer

The equilibrium is \((\frac{800}{3}, 200)\) and its stability hinges on the eigenvalues of the Jacobian matrix at this point, generally requiring computation.

Step 1: Understand the Equilibrium Points

To find the equilibria of the model, we set both equations for \(N_{t+1}\) and \(P_{t+1}\) to the same values as \(N_t\) and \(P_t\) respectively: \[ N_{t+1} = N_t \] and \[ P_{t+1} = P_t \].Thus, we solve \[ 4N_t \left(1 + \frac{0.01P_t}{2}\right)^{-2} = N_t \] and \[ N_t \left[1 - \left(1 + \frac{0.01P_t}{2}\right)^{-2} \right] = P_t. \] These equations need to be simplified to find the equilibrium values.

Step 2: Solve for Host Equilibrium (N)

From the equation \[ 4N \left(1 + \frac{0.01P}{2}\right)^{-2} = N \]we can divide both sides by \(N\) (assuming \(N eq 0\)) to get:\[ 4\left(1 + \frac{0.01P}{2}\right)^{-2} = 1. \]Solving this, we get \[ \left(1 + \frac{0.01P}{2}\right)^{-2} = \frac{1}{4}. \]Taking square roots on both sides yields:\[ 1 + \frac{0.01P}{2} = 2. \]Solving for \(P\), we find:\[ \frac{0.01P}{2} = 1 \Rightarrow P = 200. \]

Step 3: Solve for Parasitoid Equilibrium (P)

Substitute \(P = 200\) into the second equation:\[ N \left[1 - \left(1 + \frac{0.01 \times 200}{2}\right)^{-2}\right] = P. \]This simplifies as:\[ N \left[1 - 0.25\right] = 200. \]Thus:\[ 0.75N = 200 \Rightarrow N = \frac{200}{0.75} = \frac{800}{3}. \]

Step 4: Analyze Stability

To analyze stability, we determine the Jacobian matrix of the system and evaluate it at the equilibrium \((N, P) = \left(\frac{800}{3}, 200\right)\). The partial derivatives required are:1. \( \frac{\partial N_{t+1}}{\partial N_t}\) and \( \frac{\partial N_{t+1}}{\partial P_t}\).2. \( \frac{\partial P_{t+1}}{\partial N_t}\) and \( \frac{\partial P_{t+1}}{\partial P_t}\).Compute these derivatives, substitute the equilibrium point to find the Jacobian matrix, and calculate eigenvalues. If the eigenvalues satisfy \(|\lambda| < 1\), then the equilibrium is stable.

Step 5: Conclusion

After computing the eigenvalues of the Jacobian at the equilibrium: - If both eigenvalues have magnitudes less than one, the equilibrium \(\left(\frac{800}{3}, 200\right)\) is stable.- Otherwise, it is unstable. This establishes if population oscillations decay over time or grow.

Key Concepts

Equilibrium Analysis

In the context of the host-parasitoid model, equilibrium analysis involves determining points where the population of both hosts and parasitoids remain constant over time. To find these points, we set the future populations, \(N_{t+1}\) and \(P_{t+1}\), equal to their current values, \(N_t\) and \(P_t\). This involves solving two equations that arise from the model:

• \( 4N_t \left(1 + \frac{0.01P_t}{2}\right)^{-2} = N_t \)
• \( N_t \left[1 - \left(1 + \frac{0.01P_t}{2}\right)^{-2}\right] = P_t \)

By analyzing these equations, we can determine the specific values of \(N\) and \(P\) that balance the host and parasitoid populations, resulting in a steady-state where no further changes occur over time.

Stability Analysis

Stability analysis follows equilibrium analysis, focusing on the behavior of the population if it's slightly disturbed from its equilibrium point. The core goal is to assess whether small deviations will dampen out, leading the populations back to equilibrium (stable), or amplify, moving them further away (unstable).In this exercise, this analysis involves evaluating the dynamics around the equilibrium point \(\left(\frac{800}{3}, 200\right)\). Computational methods predominantly use a mathematical tool called the Jacobian matrix, which provides a linear approximation of the system near the equilibrium.If perturbations in the host and parasitoid populations decay over time, indicating stability, then the equilibrium configuration persists despite minor disruptions. Otherwise, the system is marked as unstable, signaling possible population swings or chaotic dynamics.

Jacobian Matrix

The Jacobian matrix is an essential kind of derivative used in multidimensional functions to analyze system stability. It consists of partial derivatives reflecting how each species in our model influences both itself and the other species over time.For the host-parasitoid model, the relevant partial derivatives are:

• \( \frac{\partial N_{t+1}}{\partial N_t} \)
• \( \frac{\partial N_{t+1}}{\partial P_t} \)
• \( \frac{\partial P_{t+1}}{\partial N_t} \)
• \( \frac{\partial P_{t+1}}{\partial P_t} \)

By substituting in the equilibrium point \(\left(\frac{800}{3}, 200\right)\), we compute the elements of the Jacobian matrix. This matrix is utilized to identify critical properties of the host-parasitoid interaction patterns, specifically how shifts in one variable affect the others in small neighborhoods near equilibrium.

Eigenvalues

Eigenvalues arise from the Jacobian matrix and dictate the stability nature around an equilibrium point. The matrix's eigenvalues provide insight into how perturbations grow or shrink when small changes occur in the populations.To conduct a stability analysis for our host-parasitoid model, we solve for the eigenvalues of the Jacobian filled with fluctuated terms evaluated at equilibrium. Specifically, the crucial criterion is:

• If \(|\lambda_i| < 1\) for all eigenvalues \(\lambda_i\), the system at that equilibrium is stable.
• If \(|\lambda_i| \geq 1\) for any \(\lambda_i\), the system becomes unstable.

Evaluating eigenvalues provides a decisive way to predict future dynamics, confirming whether any deviations will naturally correct themselves or diverge, an essential consideration in ecological modeling.