Consider a model of a food chain of three species. Assume that each population on its own can be modeled by logistic growth. Let the species be labeled by \(x(t), y(t)\), and \(z(t)\). Assume that population \(x\) is at the bottom of the chain. That population will be depleted by population \(y\). Population \(y\) is sustained by \(x^{\prime}\) s, but eaten by \(z^{\prime}\) s. A simple, but scaled, model for this system can be given by the system $$ \begin{aligned} x^{\prime} &=x(1-x)-x y \\ y^{\prime} &=y(1-y)+x y-y z \\ z^{\prime} &=z(1-z)+y z \end{aligned} $$ a. Find the equilibrium points of the system. b. Find the Jacobian matrix for the system and evaluate it at the equilibrium points. c. Find the eigenvalues and eigenvectors. d. Describe the solution behavior near each equilibrium point. e. Which of these equilibria are important in the study of the population model and describe the interactions of the species in the neighborhood of these point(s).

In population modeling, logistic growth describes how a population grows rapidly at first, then slows down, and finally levels off as the population reaches the carrying capacity of its environment. The carrying capacity is the maximum population size that the environment can sustain indefinitely. The logistic growth model is represented by the equation \( \frac{dP}{dt} = rP \left(1 - \frac{P}{K}\right) \), where \( P(t) \) is the population at time \( t \), \( r \) is the intrinsic growth rate, and \( K \) is the carrying capacity.

In our exercise, each species in the food chain (\( x(t), y(t), z(t) \) ) follows a similar pattern of logistic growth, subject to interaction terms that represent the predator-prey dynamics. The term \( -xy \) in the equation for \( x'(t) \) represents the decrease in population \( x \) due to predation by population \( y \) , and similar terms appear in the equations for \( y'(t) \) and \( z'(t) \) . These interactions complicate the simple logistic model, resulting in a system of differential equations that can exhibit a much richer array of dynamics, including stable and unstable equilibria.

The Jacobian matrix is a powerful tool in analyzing systems of differential equations, especially those that model nonlinear dynamics like our population model. It is a matrix of all first-order partial derivatives of a vector-valued function. In essence, it represents how a small change in inputs affects the outputs of the system.

For our system of equations, the Jacobian matrix consists of the partial derivatives of each population's growth rate with respect to each population at a given point in time. The entries in the matrix tell us how changes in one population's size could result in changes in the growth rate of another. Calculating the Jacobian matrix at the equilibrium points and examining its entries provide insight into the interaction dynamics between the populations near these points. Understanding the Jacobian matrix is crucial for stability analysis and ultimately for predicting how the species will interact and evolve over time.

Eigenvalues and eigenvectors are fundamental concepts in linear algebra that have important implications for systems of differential equations. An eigenvalue is a scalar that indicates the magnitude of stretching or compressing along the direction of its associated eigenvector during a linear transformation represented by a matrix.

In the context of our ecological model, after we have found the Jacobian matrix, we determine its eigenvalues and the corresponding eigenvectors. The eigenvalues can tell us about the stability of the equilibrium points; if they all have negative real parts, the equilibrium is stable, meaning that the population will return to this point if perturbed. On the other hand, a positive real part indicates instability. The eigenvectors provide the directions in which these dynamics occur. They can give us insight into the potential behavior of the populations as they move towards or away from equilibrium points, which aids in visualizing the population dynamics.

Stability analysis is the process of determining whether small perturbations or changes in the state of a system will decay over time, leading the system back to the equilibrium, or whether the system will diverge away from it. This aspect of analysis is crucial in understanding the long-term behavior of dynamical systems like our population model.

Using the eigenvalues obtained from the Jacobian matrix, stability analysis helps to categorize each equilibrium point as a stable node, saddle point, or unstable node. If all eigenvalues are negative, the equilibrium is stable; if all are positive, it is unstable; and if there are both positive and negative eigenvalues, the equilibrium is a saddle point. This analysis can paint a picture of the population dynamics, including which species might dominate, coexist, or go extinct based on the nature of the equilibrium points in the model.