Problem 40 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The biologically relevant equilibrium \( (N^*, P^*) = (0, 0) \) is unstable.

Step 1: Set Up Equations for Equilibrium

In equilibrium, the change in population from one generation to the next is zero. Therefore, set \( N_{t+1} = N_t = N^* \) and \( P_{t+1} = P_t = P^* \). Now the equations become: \[ N^* = 4N^* \left(1 + \frac{0.01 P^*}{2}\right)^{-2} \] \[ P^* = N^* \left[1 - \left(1 + \frac{0.01 P^*}{2}\right)^{-2}\right] \].

Step 2: Solve Host Equation for P*

Simplify the host equation, \( N^* = 4N^* \left(1 + \frac{0.01P^*}{2}\right)^{-2} \). Divide both sides by \( N^* \) (assuming \( N^* eq 0 \)): \[ 1 = 4\left(1 + \frac{0.01P^*}{2}\right)^{-2} \]. Take the reciprocal: \[ 1 = \left(\frac{1}{2}\right)^2 \left(2 + 0.01P^*\right)^2 \]. This implies \( 2 + 0.01P^* = 2 \), resulting in \( 0.01P^* = 0 \), hence \( P^* = 0 \).

Step 3: Determine the Different Equilibrium Points

From Step 2, we found \( P^* = 0 \). Substitute \( P^* = 0 \) into the parasitoid equation: \[ P^* = N^* \left[1 - \left(1 + 0\right)^{-2}\right] \Rightarrow P^* = N^*[1-1] = 0 \]. Thus, the equilibrium \( (N^*, P^*) = (0, 0) \). To find another equilibrium, consider the case \( N^* = 0 \), leading also to \( P^* = 0 \) based on the equations.

Step 4: Analyze Stability of the Equilibria

Consider \( (N^*, P^*) = (0, 0) \). Linearize the system around this point and analyze the Jacobian matrix: \[ J = \begin{bmatrix} 4 + 0.4P_t & -\frac{4N_t}{2}\left(1 + \frac{0.01 P_t}{2}\right)^{-3} \ \left(1 + \frac{0.01 P_t}{2}\right)^{-2} & N_t \end{bmatrix} \]. At \( (0,0) \), evaluate \[ J = \begin{bmatrix} 4 & 0 \ 1 & 0 \end{bmatrix} \]. The eigenvalues of this matrix, obtained from \( \det(J - \lambda I) = 0 \), indicate instability as they are \( \lambda = 4, \lambda = 0 \). Hence, \( (0,0) \) is unstable. Further analysis at \( N^* > 0 \), \( P^* = 0 \), similarly reveals instability due to positive eigenvalues.

Key Concepts

Equilibrium Analysis

Equilibrium analysis in ecological modeling is about finding points where species populations remain constant over time. In the host-parasitoid model we consider, equilibrium means the populations of both the host and the parasitoid do not change from one time step to the next. Mathematically, this is expressed as setting the population equations equal to themselves across time steps. Here, it is when \( N_{t+1} = N_t = N^* \) and \( P_{t+1} = P_t = P^* \).
To find equilibria, we solve the given equations to identify values of \( N^* \) and \( P^* \) that satisfy this constancy condition. In our specific equations:

• For hosts: \( N^* = 4N^* \left(1 + \frac{0.01 P^*}{2}\right)^{-2} \)
• For parasitoids: \( P^* = N^*\left[1 - \left(1 + \frac{0.01 P^*}{2}\right)^{-2}\right] \)

Solving these reveals that both \( (N^*, P^*) = (0, 0) \) and \( (N^*, P^*) = (0, \, P^*) \) are potential equilibria, but the latter still leads to \( P^* = 0 \), reinforcing the point at \( (0, 0) \). Hence, biological relevance limits us typically to this simplest form or trivial equilibria in populations modeling.

Stability Analysis

Stability analysis helps us understand if small changes or perturbations in population sizes at equilibrium will return to equilibrium or deviate away from it. In mathematical models like the host-parasitoid model, this is examined via the Jacobian matrix and its eigenvalues at the equilibrium points.
For the equilibrium \( (N^*, P^*) = (0, 0) \), we linearize the system by calculating the Jacobian matrix:

• \( J = \begin{bmatrix} 4 + 0.4P_t & -\frac{4N_t}{2}\left(1 + \frac{0.01 P_t}{2}\right)^{-3} \ \left(1 + \frac{0.01 P_t}{2}\right)^{-2} & N_t \end{bmatrix} \)

Evaluating this at \( (0, 0) \) gives \( J = \begin{bmatrix} 4 & 0 \ 1 & 0 \end{bmatrix} \).
Eigenvalues of this matrix, found from \( \det(J - \lambda I) = 0 \), drives how perturbations evolve. Here, the eigenvalues are \( \lambda = 4 \) and \( \lambda = 0 \). A positive eigenvalue (\( \lambda = 4 \)) signals that the equilibrium is unstable. This indicates that any small population changes will likely grow rather than settle back to the equilibrium point.

Ecological Modeling

Ecological modeling is a powerful method to understand interactions and dynamics between species, such as hosts and parasitoids. These models use mathematical equations to represent biological relationships and predict population behaviors through time.
The model examined incorporates a negative binomial response function to describe changes in host \( (N_t) \) and parasitoid \( (P_t) \) populations. Parameters like carrying capacity and reproduction rate are vital. In our example, parameters within the equations are used to gauge how the host population influences parasitoid success rates. The model helps in assessing functional responses like increased parasitism as host density increases, which in this case is modeled by the \( \left(1 + \frac{0.01P^*}{2}\right)^{-2} \) term in the equations.

• The complexity of the model caters to realistic population dynamics and interspecies impact assessment.
• This predictive insight assists in ecological management and conservation efforts by projecting potential future population trajectories.

Overall, ecological modeling like this informs us how small changes in parameters or initial conditions can affect the coexistence or extinction of species, providing vital knowledge for ecological balance and species management.