Problem 37 - Calculus for Biology and Medicine

Question

Denote by \(x_{1}(t)\) the number of juveniles, and by \(x_{2}(t)\) the number of adults, at time \(t\). Assume that \(x_{1}(t)\) and \(x_{2}(t)\) evolve according to $$ \begin{array}{l} x_{1}(t+1)=x_{2}(t) \\ x_{2}(t+1)=\frac{1}{2} x_{1}(t)+r x_{2}(t)-x_{2}^{2}(t) \end{array} $$ (a) Show that if \(r>1 / 2\), there exists an equilibrium \(\left[\begin{array}{c}x_{1}^{*} \\ x_{2}^{*}\end{array}\right]\) with \(x_{1}^{*}>0\) and \(x_{2}^{*}>0\). Find \(x_{1}^{*}\) and \(x_{2}^{*}\). (b) Determine the stability of the equilibrium found in (a) when \(r>1 / 2 .\)

Step-by-Step Solution

Answer

If \(r > 1/2\), the equilibrium \(x_1^* = x_2^* = r - 1/2\) exists. Stability depends on the eigenvalues of the Jacobian matrix.

Step 1: Define Equilibrium Conditions

At equilibrium, the populations do not change over time. Thus, we set \(x_1(t+1) = x_1(t)\) and \(x_2(t+1) = x_2(t)\). This yields the equations:1. \(x_1^* = x_2^*\)2. \(x_2^* = \frac{1}{2}x_1^* + rx_2^* - (x_2^*)^2\) We will solve these to find \(x_1^*\) and \(x_2^*\).

Step 2: Solve the Equilibrium Equations

From step 1, we have \(x_1^* = x_2^*\). Substituting into the second equation:\[ x_2^* = \frac{1}{2}x_2^* + rx_2^* - (x_2^*)^2 \]Rearranging gives:\[ 0 = -\frac{1}{2}x_2^* + rx_2^* - (x_2^*)^2 \]This can be rewritten as:\[ (x_2^*)^2 - \left(r - \frac{1}{2}\right)x_2^* = 0 \]Factoring out \(x_2^*\), we obtain:\[ x_2^*(x_2^* - (r - \frac{1}{2})) = 0 \]

Step 3: Find Positive Equilibrium

The equation \( x_2^*(x_2^* - (r - \frac{1}{2})) = 0 \) gives solutions \( x_2^* = 0 \) or \( x_2^* = r - \frac{1}{2} \). For an equilibrium with \( x_2^* > 0 \), we need the second solution. Thus, if \( r > \frac{1}{2} \), then \( x_2^* = r - \frac{1}{2} \) is positive.Since \( x_1^* = x_2^* \), we have \( x_1^* = x_2^* = r - \frac{1}{2} \).

Step 4: Determine Stability of Equilibrium

To determine stability, consider the Jacobian matrix of the system:\[ J = \begin{bmatrix} 0 & 1 \ \frac{1}{2} & r - 2x_2^* \end{bmatrix} \]At equilibrium \( x_2^* = r - \frac{1}{2} \), compute the Jacobian:\[ J = \begin{bmatrix} 0 & 1 \ \frac{1}{2} & -\left(\frac{1}{2} - r\right) \end{bmatrix} \]The eigenvalues are found by solving the characteristic equation:\[ \lambda^2 - \left( -\frac{1}{2} + r \right)\lambda - \frac{1}{2} = 0 \]The solutions to this quadratic will determine stability.

Key Concepts

Stability Analysis

Stability Analysis is a crucial aspect of understanding equilibrium in systems like population dynamics. It involves examining whether small changes to a system cause it to return to, or deviate further from, an equilibrium state. Stability is determined by looking at an equilibrium point and assessing if small perturbations around it damp out over time—all thanks to the system's intrinsic properties.

The idea is that stable equilibrium means the population returns to balance after minor disturbances. In our example, once we solve for the equilibria with conditions such as stability, we evaluate the Jacobian matrix at the equilibrium point. The Jacobian matrix assists in linearizing the system around the equilibrium, showcasing potential perturbations.

The eigenvalues of the Jacobian determine stability:

• If all eigenvalues have negative real parts, the system is stable; disturbances decay over time.
• Non-negative real parts mean the system is unstable; disturbances grow.

In conclusion, stability analysis gives insights into the behavior of populations over time, indicating whether they tend to settle into a steady state or not.

Difference Equations

Difference equations play a crucial role in modeling dynamic systems that evolve in discrete time steps, particularly in population dynamics. They provide a framework to predict future states based on current conditions. Essentially, a difference equation relates the value of a variable at one time point to its previous values.

For our given system, two difference equations define the evolution of juveniles and adults. We have

• Equation 1: \( x_{1}(t+1) = x_{2}(t) \)
• Equation 2: \( x_{2}(t+1) = \frac{1}{2} x_{1}(t) + r x_{2}(t) - x_{2}^{2}(t) \)

These equations reflect the dependency of future population distributions on current ones. Solving these equations under equilibrium conditions (where populations don't change over time) provides vital equilibrium states. Such states are integral in determining stability and long-term predictions.

Difference equations thus serve as the backbone for analyzing how systems change over discrete-time steps, offering a structured approach to forecasting and stability analysis.

Population Dynamics

Population dynamics explores how biological populations change over time and the processes affecting these changes. This field is essential in understanding growth patterns, fluctuations, and long-term sustainability of species in ecosystems.

In our exercise, population dynamics is explored through a model comprising juveniles and adults, each governed by its specific rule of progression. The movement from juveniles to adults and changes in adult numbers exhibit how interdependent factors influence population changes over time.

Key elements include:

• Birth rates and transition from juveniles to adults
• Adult survival rates influenced by parameters like \( r \)

The sophisticated equation for adults, \( x_{2}(t+1) = \frac{1}{2} x_{1}(t) + r x_{2}(t) - x_{2}^{2}(t) \), highlights competitive factors among adults that affect overall population sustainability.

Studying population dynamics with such models helps ecologists and biologists make informed decisions about conservation efforts and predict ecological outcomes. Understanding these dynamics assists in identifying stable states, potential population booms or crashes, and the influence of interspecies interactions.