Problem 89 - Calculus for Biology and Medicine

Question

Suppose that \(N(t)\) denotes the size of a population at time \(t\). The population evolves according to the logistic equation, but in addition, predation reduces the size of the population so that the rate of change is given by $$ \frac{d N}{d t}=g(N) $$ where $$ g(N)=N\left(1-\frac{N}{50}\right)-\frac{9 N}{5+N} $$ The first term on the right-hand side describes the logistic growth; the second term describes the effect of predation. (a) Make the vector field plot for this differential equation. (b) Find all equilibria of \((8.38)\). (c) Use your vector field plot in (a) to determine the stability of the equilibria you found in (b). (d) Repeat your analysis from part (c) but now use the method of eigenvalues to determine the stability of the equilibria you found in (b).

Step-by-Step Solution

Answer

The equilibria occur at \( N=0 \) and other values found from solving \( N \left( 1 - \frac{N}{50} \right) = \frac{9N}{5+N} \). Stability can be analyzed using the vector field directions and eigenvalues of the system.

Step 1: Define Vector Field Elements

To create a vector field plot for the differential equation \( \frac{dN}{dt} = g(N) \), identify \( g(N) = N \left( 1 - \frac{N}{50} \right) - \frac{9N}{5+N} \). This forms the basis for calculating the direction and magnitude of change at various population sizes.

Step 2: Identify Equilibria Points

Set \( g(N) = 0 \) to find the equilibria points. Solve \( N \left( 1 - \frac{N}{50} \right) = \frac{9N}{5+N} \). Simplify to find \( N = 0 \) and other non-zero solutions for \( N \).

Step 3: Simplify the Equilibria Equation

Rearrange the equation \( N - \frac{N^2}{50} = \frac{9N}{5+N} \) to form a polynomial or simpler expression. This involves multiplying through \( (5+N) \) and rearranging terms to find the non-zero equilibria.

Step 4: Solve for Non-zero Equilibria

Solve the polynomial by factoring or using suitable methods to find the values of \( N \) where the population size is in equilibrium, i.e., where the rate of change is zero.

Step 5: Sketch Vector Field Plot

Plot a vector field for \( N \) against \( g(N) \) showing arrows that represent the direction and steepness (rate) of population change based on different values of \( N \). Draw arrows pointing right or left based on positive or negative values of \( g(N) \).

Step 6: Determine Stability from Plot

Using the vector field plot, observe if arrows point towards an equilibrium (indicating stability) or away (indicating instability). Evaluate how the direction of vectors changes around each equilibrium.

Step 7: Derive Jacobian for Stability Analysis

Find the derivative \( \frac{dg}{dN} \) to use in the Jacobian (eigenvalue method). This derivation involves finding the rate of change of \( g(N) \) with respect to \( N \).

Step 8: Calculate Eigenvalues for Stability

Evaluate the stability at each equilibrium by substituting into the derivative to get the eigenvalues. If the eigenvalue is negative, the equilibrium is stable; if positive, it's unstable.

Step 9: Conclude Stability Analysis

Combine results from the vector field plot and eigenvalue calculations to conclude the stability of equilibria points. Use these to summarize the overall behavior of the population growth model.

Key Concepts

Population Dynamics

Population dynamics is the study of how populations change over time. In the context of the logistic growth model, it is particularly interesting because it includes both growth and limiting factors.
The logistic model assumes populations grow exponentially when resources are unlimited, but this growth is tempered as the population nears a carrying capacity due to resource limitations. This behavior is encapsulated by the term \( N \left(1 - \frac{N}{K}\right) \), where \( N \) is the current population size and \( K \) is the carrying capacity.
This growth is further influenced by predation, as seen in the exercise through the term \( -\frac{9N}{5+N} \). This represents the negative effect of predation on the population, reducing its growth rate. Ultimately, population dynamics involves understanding these interactions and predicting population changes over time based on these biological and environmental influences.

Differential Equations

Differential equations form the backbone of modeling population dynamics in the logistic growth context. A differential equation describes how a function changes, making it perfect for modeling population growth over time.
In this case, the differential equation is given as \( \frac{dN}{dt} = g(N) \), which defines the rate of change of the population \( N \) with respect to time \( t \). By examining how \( g(N) = N\left(1-\frac{N}{50}\right)-\frac{9N}{5+N} \) behaves, we can determine how the population size changes as time progresses.
Solving this differential equation involves finding equilibrium points where \( \frac{dN}{dt} = 0 \), as these are the points where the population size does not change. These equations require tools like algebraic manipulation, factoring, or numerical methods for solutions, especially when complex interactions like predation are involved.

Equilibria Stability

Understanding the stability of equilibria is crucial for predicting the long-term behavior of a population. An equilibrium occurs when the population size remains constant, meaning \( \frac{dN}{dt} = 0 \).
To determine the stability of these equilibria, we can use the vector field plot, which visualizes how the population size will change from various points. If vectors around an equilibrium point point towards it, the equilibrium is stable; if they point away, it's unstable.
The Jacobian matrix and the eigenvalue method offer a more mathematical approach. By finding \( \frac{dg}{dN} \), the derivative of the growth rate with respect to population size, and calculating its eigenvalues, we can predict stability analytically.

• A negative eigenvalue indicates a stable equilibrium.
• A positive eigenvalue suggests instability.

Together, these methods give a comprehensive view of how a population might behave around its equilibria.