Problem 20 - Calculus for Biology and Medicine

Question

A generalization of the Beverton-Holt model for population growth was created by Hassell (1975). Under Hassell's model the population \(N_{t}\) at discrete times \(t=0,1,2, \ldots\) is modeled by a recurrence equation: $$N_{t+1}=\frac{R_{0} N_{t}}{\left(1+a N_{t}\right)^{c}}$$ where \(R_{0}, a\), and \(c\) are all positive constants. (a) Explain why you would expect that \(R_{0}>1\). (b) Assume that \(c=2, R_{0}=9\), and \(a=\frac{1}{10} .\) Find all possible equilibria of the system. (c) Use the stability criterion for equilibria to determine which, if any, of the equilibria of the recursion relation are stable.

Step-by-Step Solution

Answer

(a) $R_0 > 1$ ensures growth without limits. (b) Equilibria: $N = 0, 20$. (c) $N = 0$ is unstable, $N = 20$ is stable.

Step 1: Understanding the Role of $R_0$

In the model, $R_0$ represents the growth rate of the population without any limiting factors. We expect $R_0 > 1$ because it indicates that the population will grow if not constrained by any environmental factors. If $R_0  1$, the population would remain constant or decrease even without constraints.

Step 2: Setting $N_{t+1} = N_t$ for Equilibrium

To find the equilibria, set \(N_{t+1} = N_t = N_{eq}\) in the recurrence relation:\[N_{eq} = \frac{9 N_{eq}}{\left(1 + \frac{1}{10} N_{eq}\right)^2}\]Simplifying this equation gives the condition for equilibrium.

Step 3: Solving for Equilibrium

Cancel \(N_{eq}\) (assuming \(N_{eq} eq 0\)) and solve:\[1 = \frac{9}{(1 + \frac{1}{10} N_{eq})^2}\]Let \(x = 1 + \frac{1}{10} N_{eq}\), then: \[x^2 = 9\]Solving gives \(x = 3\) (we discard \(x = -3\) due to a positive population). Thus, \(\frac{1}{10} N_{eq} = 2\), so \(N_{eq} = 20\).

Step 4: Equilibrium Solutions

Therefore, the possible equilibria are $N_{eq} = 0$ and $N_{eq} = 20$.

Step 5: Stability Criterion Evaluation

The stability criterion involves evaluating the derivative of the function \(f(N) = \frac{9N}{(1+\frac{1}{10}N)^2}\) at the equilibria.Find the derivative \(f'(N)\):\[f'(N) = \frac{d}{dN}\left(\frac{9N}{(1+\frac{1}{10}N)^2}\right)\]Compute \(f'(0)\) and \(f'(20)\) to analyze stability: \[f'(0) = 9\], which is greater than 1 (unstable).For \(N_{eq} = 20\), calculate \(f'(20)\). If \(|f'(20)| < 1\), \(N = 20\) is stable.

Step 6: Evaluate Derivative at $N_{eq} = 20$

Calculate \(f'(20)\):The derivative simplifies to:\[f'(N) = \frac{9}{(1+\frac{1}{10}N)^2} - \frac{18N}{(1+\frac{1}{10}N)^3}\]Evaluate at \(N = 20\), \(f'(20) < 1\) so it is stable.

Key Concepts

Beverton-Holt model

The Beverton-Holt model is a classic population growth model used to understand how populations grow under limited resources. It assumes that the environment offers only finite resources, so the population cannot grow indefinitely. The model is often represented as a recurrence equation.
The general form is:

• \( N_{t+1} = \frac{R_0 N_t}{1 + a N_t} \)

Here, \(N_t\) is the population size at time \(t\), \(R_0\) is the intrinsic growth rate, and \(a\) is a constant related to the environment's carrying capacity. When \(R_0 > 1\), it indicates that the population grows while \(R_0 \leq 1\) suggests stagnation or decline without external constraints.
This model is significant because it illustrates how population growth slows as it approaches the carrying capacity, achieving a natural equilibrium.

Hassell's model

Hassell’s model extends the Beverton-Holt model to account for more complex interactions within a population, particularly when competition increases sharply as population density goes up. It is shown through the equation:

• \( N_{t+1} = \frac{R_0 N_t}{(1 + a N_t)^c} \)

In this equation, \(c\) is a positive constant that influences how significantly competition affects the population.
When \(c = 1\), Hassell’s model simplifies to the Beverton-Holt model. As \(c\) increases, competition effects intensify, making higher population densities more unsustainable.
The parameters \(R_0\), \(a\), and \(c\) together determine the shape and dynamics of the population's growth curve. With \(R_0 > 1\), the population can grow; however, the model suggests that increased density-dependent competition eventually limits growth.

Equilibrium stability

Equilibrium in population models refers to a state where the population size remains constant over time. It occurs when the population at time \(t\) equals the population at time \(t+1\). In equations like Hassell's model, reaching equilibrium requires:

• \( N_{t+1} = N_t = N_{eq} \)

To explore stability, one examines whether small perturbations around \(N_{eq}\) will return to equilibrium or move away.
If a population is perturbed slightly and returns to equilibrium, it is stable. If it moves away, the equilibrium is unstable. Stability is often determined by evaluating the derivative of the function describing population growth at \(N_{eq}\).
For Hassell’s model, the equilibrium solutions might be \(N_{eq} = 0\) or some positive \(N_{eq}\), depending on parameter values. Examining the derivative helps determine which equilibrium points are stable.

Recurrence equation

A recurrence equation is a key mathematical concept in modeling population dynamics.

• It defines the next value in a sequence based on the preceding one.
• The general form is: \( N_{t+1} = f(N_t) \)

The equation models how a process evolves over discrete time steps \(t = 0, 1, 2, \ldots\). In population models, recurrence equations predict future population sizes using current sizes and model parameters.
Hassell's and the Beverton-Holt models both use recurrence equations to simulate population dynamics, accounting for growth and environmental limits.
The solutions to these equations help in understanding long-term behaviors like equilibrium and stability of the population under various conditions.