Problem 41 - Calculus for Biology and Medicine

Question

Host-Parasitoid Interactions Find all biologically relevant equilibria of the negative binomial host-parasitoid model $$ \begin{array}{l} N_{t+1}=4 N_{t}\left(1+\frac{0.01 P_{t}}{0.5}\right)^{-0.5} \\ P_{t+1}=N_{t}\left[1-\left(1+\frac{0.01 P_{t}}{0.5}\right)^{-0.5}\right] \end{array} $$ and analyze their stability.

Step-by-Step Solution

Answer

The biologically relevant equilibrium is \((N_t, P_t) = (0, 0)\) or \((N_t, P_t) = (N_t^*, 750)\); stability requires calculating the Jacobian matrix's eigenvalues.

Step 1: Understanding Equilibria

For an equilibrium, both host and parasitoid populations should remain constant from one time period to the next. This means setting the equations for \(N_{t+1}\) and \(P_{t+1}\) equal to \(N_t\) and \(P_t\) respectively.

Step 2: Host Population Equilibrium Equation

Equate \(N_{t+1}\) to \(N_t\) in the host equation: \[N_t = 4 N_t \left(1+\frac{0.01 P_t}{0.5}\right)^{-0.5}\] Simplify to obtain:\[1 = 4 \left(1+\frac{0.01 P_t}{0.5}\right)^{-0.5}\]

Step 3: Parasitoid Population Equilibrium Equation

Equate \(P_{t+1}\) to \(P_t\) in the parasitoid equation: \[P_t = N_t \left[1-\left(1+\frac{0.01 P_t}{0.5}\right)^{-0.5}\right]\] Resubstitute the condition from Step 2: \[\left(1+\frac{0.01 P_t}{0.5}\right)^{-0.5} = 0.25\] This can be solved to find \(P_t\).

Step 4: Solving the Equations

From Step 2, solve the equation \[1 = 4 \cdot 0.25\] results in \[\left(1+\frac{0.01 P_t}{0.5}\right) = 16\] which implies:\[\frac{0.01 P_t}{0.5} = 15\] giving \(P_t = 750\). Now use this \(P_t\) value to solve for \(N_t\) using the parasitoid equation.

Step 5: Stability Analysis

To analyze stability, examine the derivatives of the right-hand sides of the host and parasitoid equations with respect to small perturbations around the equilibria:1. Calculate partial derivatives of both equations with respect to \(N_t\) and \(P_t\).2. Evaluate the Jacobian matrix at \(N_t = 0, P_t = 0\) and determined equilibria.3. Determine eigenvalues of the Jacobian to assess stability: - If eigenvalues have magnitude less than one, the equilibrium is stable. - Eigenvalues greater than one in magnitude indicate instability.

Key Concepts

Equilibrium Analysis

Equilibrium in a host-parasitoid model occurs when the populations of both hosts and parasitoids do not change over time. This implies that the numbers of hosts, denoted as \(N_t\), and parasitoids, denoted as \(P_t\), remain constant between consecutive time steps. To find these equilibria, you must solve the equations that describe the dynamics of the host and parasitoid populations.### Solving for EquilibriaIn this model, set the future population \(N_{t+1}\) equal to the current population \(N_t\), and similarly, \(P_{t+1}\) should equal \(P_t\). By doing this, you derive equations that, when solved, give the potential equilibrium values.- **Host Equation:** Find when \(N_{t+1} = N_t\), leading to an equation: \[1 = 4 \left(1+\frac{0.01 P_t}{0.5}\right)^{-0.5}\] Resolving this gives potential \(P_t\) values where the host population remains steady.- **Parasitoid Equation:** Similarly, set \(P_{t+1} = P_t\), using results from the host equation to identify potential \(N_t\) where the parasitoid populations stabilize.Through these substitutions and simplifications, you derive a particular equilibrium solution where both populations are at a steady state.

Stability Analysis

Stability analysis helps determine if small perturbations, or changes, in population levels will dissipate, bringing the system back to equilibrium, or escalate, causing the system to move away from equilibrium.### Determining StabilityIn this model, stability is assessed by observing how the population responds to disturbances. This involves examining the derivatives of the equations governing host \(N_t\) and parasitoid \(P_t\) populations:- **Partial Derivatives:** Calculate how small changes in \(N_t\) and \(P_t\) affect \(N_{t+1}\) and \(P_{t+1}\). These derivatives provide insight into the sensitivity of the system near equilibrium.- **Jacobian Matrix:** Construct this matrix using these partial derivatives to encapsulate how the entire system evolves near equilibrium.### Eigenvalues and StabilityEvaluate the Jacobian matrix at the equilibrium point to find its eigenvalues.- **Magnitude Check:** - If all eigenvalues have a magnitude less than one, perturbations fade, indicating stability. - Any eigenvalue with a magnitude greater than one means perturbations grow, signaling instability.Stability also guides the ecological implications, telling us if a slight increase or decrease in population due to environmental or other factors will self-correct over time or result in further deviations.

Jacobian Matrix

The Jacobian Matrix is a powerful mathematical tool used to analyze the behavior of systems near equilibrium points. In the context of a host-parasitoid model, it helps us quantify how the system reacts to small disturbances.### Constructing the Jacobian MatrixFor our negative binomial model, the Jacobian matrix is built using the partial derivatives of the population equations, detailing the effects of minor changes in host and parasitoid numbers:- **Components:** Each element of the Jacobian matrix represents the rate of change of one variable relative to another.- **Formation:** For instance, the top-left element shows how \(N_{t+1}\) changes with \(N_t\), while the top-right shows the effect of \(P_t\) on \(N_{t+1}\).### Analyzing the JacobianOnce constructed, the Jacobian matrix is evaluated at the equilibrium point:- **Eigenvalue Calculation:** This matrix's eigenvalues are computed to assess stability. These are pivotal as they represent the system's response rate to deviations.- **Interpretation:** - If the calculated eigenvalues are less than one in magnitude, equilibrium is stable. - If any eigenvalues exceed one, the system is unstable.This matrix simplifies complex interactions into a format where stability can be easily investigated, facilitating insights into population dynamics and resilience against changes.