Problem 24 - Calculus for Biology and Medicine

Question

Consider density-dependent population growth models of the form $$ N_{t+1}=R\left(N_{t}\right) N_{t} $$ The function \(R(N)\) describes the per capita growth. Various forms have been considered. For each function \(R(N)\), find all nontrivial fixed points \(N^{*}\) (i.e, \(N^{*}>0\) ) and determine the stability as a function of the parameter values. We assume that the function parameters are \(r>0, K>0\), and \(\gamma>1 .\) for \(t=0,1,2, \ldots\) \(R(N)=\frac{r}{1+N / K}\)

Step-by-Step Solution

Answer

The nontrivial fixed point is \(N^* = K(r-1)\) and it is stable for \(r > 1\).

Step 1: Define the Fixed Point Condition

A fixed point is where the population does not change over time, so we set the equation \[ N_{t+1} = N_t = N^* \] Thus, the fixed point equation becomes \[ R(N^*)N^* = N^* \] which simplifies to \[ \frac{rN^*}{1+N^*/K} = N^* \] provided that \(N^* eq 0\).

Step 2: Simplify the Fixed Point Equation

To solve for \( N^* \), first divide both sides of the equation by \( N^* \) (assuming it is non-zero): \[ \frac{r}{1 + N^*/K} = 1 \] Multiply both sides by \(1 + N^*/K\) to clear the fraction: \[ r = 1 + N^*/K \]

Step 3: Solve for the Nontrivial Fixed Point

Rearrange the equation to solve for \( N^* \): \[ N^*/K = r - 1 \]Thus, \[ N^* = K(r - 1) \] This is the nontrivial fixed point \( N^* \). Since \(r > 1\), \( N^* > 0 \) is assured.

Step 4: Determine Stability of the Fixed Point

For stability, consider the derivative of the growth function. The Jacobian or change around the fixed point is given by \[ f'(N) = R(N) + R'(N)N \] Compute \( R'(N) = -\frac{rK^{-1}}{(1 + N/K)^2} \) and substitute in the fixed point: \[ f'(N^*) = \frac{r}{1 + N^*/K} - \frac{rN^*/K}{(1 + N^*/K)^2} \] Stability occurs when \(|f'(N^*)| < 1\). After simplification by substituting \( N^* = K(r-1) \), it turns out that as long as \(r > 1\), the fixed point remains stable.

Key Concepts

Fixed Points and Their Importance in Population Models

Fixed points are a central concept in analyzing population growth models. Essentially, a fixed point is a value or state where a system can remain unchanged over time. This means that if a population starts at this point, it will stay there unless perturbed by external factors. In the context of our population model, it's where the population remains stable given the current conditions.
Think of it like a balance point on a seesaw; when reached, no net change occurs. In mathematical terms, for our model described by the equation \( N_{t+1} = R(N_{t}) N_{t} \), a fixed point \( N^* \) satisfies the equation \( R(N^*)N^* = N^* \).
When we derive fixed points, we're looking for nontrivial solutions where the population is not zero. Mathematically, this leads us to solve the equation \[ \frac{r}{1 + N^*/K} = 1 \] to find \( N^* = K(r - 1) \). This solution demonstrates how parameters like \( r \) (growth rate) and \( K \) (carrying capacity) influence population stability.

Stability Analysis of Fixed Points

Stability analysis helps us understand whether a fixed point is attractive (stable) or repulsive (unstable). In simple terms, a stable fixed point means that if the population slightly deviates from this point, it will return to it over time.
To determine stability, we examine the derivative of the function at the fixed point, often expressed in terms of a Jacobian for more complex models. Here, we use the derivative \( f'(N) = R(N) + R'(N)N \).
The condition for stability is that the absolute value of this derivative at the fixed point should be less than 1, i.e., \(|f'(N^*)| < 1\). This ensures that small perturbations will not lead to dramatic changes in population size. In our case, substituting \( N^* = K(r-1) \) into the derivative reveals that when \( r > 1 \), the fixed point remains stable.

Understanding Per Capita Growth

Per capita growth refers to the growth rate per individual in the population. It's an essential concept in population dynamics as it helps to understand how quickly the population is increasing or decreasing per member.
For our model, per capita growth is encapsulated in the function \( R(N) = \frac{r}{1+N/K} \). This function indicates how growth rate is affected by population size \( N \) in relation to carrying capacity \( K \).
As the population size \( N \) approaches the carrying capacity \( K \), the denominator increases, causing the per capita growth rate \( \frac{r}{1+N/K} \) to decrease. This reflects density dependence, a scenario where growth slows down as resources become limited. Therefore, the model realistically reflects situations where an environment can only sustain a certain number of individuals before growth slows or stops entirely. This realistic depiction is crucial for ecologists and biologists trying to predict changes in wildlife populations or manage species conservation effectively.