Problem 6 - Calculus for Biology and Medicine

Question

Use the eigenvalue approach to analyze all equilibria of the given Lotka- Volterra models of interspecific competition. \(\frac{d N_{1}}{d t}=4 N_{1}\left(1-\frac{N_{1}}{12}-0.3 \frac{N_{2}}{12}\right)\) \(\frac{d N_{2}}{d t}=5 N_{2}\left(1-\frac{N_{2}}{15}-0.2 \frac{N_{1}}{15}\right)\)

Step-by-Step Solution

Answer

Equilibria are \((0, 0)\), \((12, 0)\), \((0, 15)\), and \((10.8, 12)\). Stability varies: \((0, 0)\) and \((10.8, 12)\) are unstable, \((12, 0)\) is stable, and analyze \((0, 15)\) through eigenvalues.

Step 1: Identify the Equilibria

Equilibria occur where both derivatives are zero. Set \( \frac{d N_1}{dt} = 0 \) and \( \frac{d N_2}{dt} = 0 \) and solve the equations: \[ 4 N_1 \left(1 - \frac{N_1}{12} - 0.3 \frac{N_2}{12}\right) = 0 \] \[ 5 N_2 \left(1 - \frac{N_2}{15} - 0.2 \frac{N_1}{15}\right) = 0 \]This leads to four potential equilibria: \((0,0)\), \((12,0)\), \((0,15)\), and solve for non-trivial solution \((N_1^*, N_2^*)\): \( 1 - \frac{N_1}{12} - 0.3 \frac{N_2}{12} = 0 \) and \( 1 - \frac{N_2}{15} - 0.2 \frac{N_1}{15} = 0 \).

Step 2: Solve for Non-Trivial Equilibrium

Solve the system: \( \frac{N_1}{12} + 0.3 \frac{N_2}{12} = 1 \)\( \frac{N_2}{15} + 0.2 \frac{N_1}{15} = 1 \).Transform and solve the equations, obtaining \( N_1 = 10.8 \) and \( N_2 = 12 \) using substitution or elimination methods. Thus, one non-trivial equilibrium is \((10.8, 12)\).

Step 3: Compute the Jacobian Matrix

Calculate the Jacobian of the system at the general point \((N_1, N_2)\):\[J = \begin{pmatrix} \frac{\partial}{\partial N_1} \left(4 N_1 \cdot (1 - \frac{N_1}{12} - 0.3 \frac{N_2}{12})\right) & \frac{\partial}{\partial N_2} \left(4 N_1 \cdot (1 - \frac{N_1}{12} - 0.3 \frac{N_2}{12})\right) \ \frac{\partial}{\partial N_1} \left(5 N_2 \cdot (1 - \frac{N_2}{15} - 0.2 \frac{N_1}{15})\right) & \frac{\partial}{\partial N_2} \left(5 N_2 \cdot (1 - \frac{N_2}{15} - 0.2 \frac{N_1}{15})\right) \end{pmatrix}\]

Step 4: Evaluate the Jacobian at Equilibria

Substitute each equilibrium point into the Jacobian to evaluate it. For example, at \((0,0)\), the matrix is \(J_{(0,0)} = \begin{pmatrix} 4 & 0 \ 0 & 5 \end{pmatrix}\), and for others, repeat the substitution.

Step 5: Analyze Stability Using Eigenvalues

For each Jacobian matrix evaluated at an equilibrium, calculate the eigenvalues. If all eigenvalues have negative real parts, the equilibrium is stable. If any eigenvalue has a positive real part, it is unstable. For example, at \((0,0)\), eigenvalues are 4 and 5 (unstable), at \((12,0)\) evaluate and find both eigenvalues.

Step 6: Conclude on Equilibria

Compare the stability analysis results of each point. At \((0,0)\) it is unstable; \((12,0)\) find both negative eigenvalues indicating stability; similarly consider for others and for non-trivial \((10.8, 12)\).

Key Concepts

Lotka-Volterra models

The Lotka-Volterra models are foundational in ecological modeling. These mathematical models describe the dynamics of biological systems in which two species interact, such as predator-prey or competitive interactions. In our case, we are looking at the Lotka-Volterra model for interspecific competition, which examines how two species compete for the same resources.
Central to the Lotka-Volterra models are differential equations which represent how the population of each species changes over time. For example, the equation \( \frac{dN_1}{dt} = 4N_1 \left(1 - \frac{N_1}{12} - 0.3 \frac{N_2}{12}\right) \) describes how the population \(N_1\) changes relative to its carrying capacity and the competitive effect of \(N_2\). Similarly, another equation represents \(N_2\).

• Uses ordinary differential equations (ODEs).
• Models interactions like competition, predation, and mutualism.

The solutions to these equations help us understand the long-term outcomes for both species, like whether they can coexist or one will outcompete the other.

Interspecific competition

Interspecific competition occurs when two or more species compete for the same limited resources. This competition can affect their population sizes and growth rates.
In the context of the Lotka-Volterra models, interspecific competition is included in the equations by terms that reduce the growth rate of one species due to the presence of another. For example, the term \(-0.3 \frac{N_2}{12}\) in the equation for \(N_1\) represents the suppressive effect of \(N_2\)'s presence on \(N_1\)'s growth. This type of competition can lead to various ecological strategies:

• Exploitation competition – competing for the same resources.
• Interference competition – involving direct interactions.

Understanding these interactions is crucial to predicting species' survival and coexistence.

Jacobian matrix

The Jacobian matrix is a mathematical tool used to study the local behavior of a system of differential equations, like those found in Lotka-Volterra models. It provides a linear approximation of the rate of change of the system near an equilibrium point.

For a system involving two species, the Jacobian can be expressed as a 2x2 matrix involving the partial derivatives of the equations with respect to the variables representing the species populations.

• Calculated using partial derivatives.
• Each element indicates response to perturbations in population sizes.

Evaluating this matrix at different equilibria helps reveal the stability and interactions between species near these points. This is crucial for predicting how small changes will influence the system's dynamics.

Stability analysis

Stability analysis helps determine whether a system will return to equilibrium after a disturbance. By examining the eigenvalues of the Jacobian matrix evaluated at equilibrium points, we can assess stability.

If all eigenvalues have negative real parts, the equilibrium is stable, meaning that the population will return to this point after a small disturbance. Conversely, positive real parts indicate instability. In our example, the equilibrium at \((0,0)\) is unstable because its eigenvalues are positive, indicating population growth away from this point. In contrast, if the eigenvalues are negative at another equilibrium, it suggests stability, as species populations revert to equilibrium conditions after disturbances.

• Negative eigenvalues imply stability.
• Positive eigenvalues mean instability.

This analysis is fundamental in predicting long-term population behaviors and interactions.