Problem 25 - 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 \(t=0,1,2, \ldots\) \(R(N)=e^{r(1-N / K)}\)

Step-by-Step Solution

Answer

The fixed point is \(N^* = K\); stable if \(0 < r < 2\), unstable if \(r > 2\).

Step 1: Define Nontrivial Fixed Points

A fixed point \(N^*\) is defined as a point where the population \(N_{t+1}\) equals \(N_t\). Therefore, for a fixed point, we have the equation: \[ N^* = R(N^*) N^* \] Substituting \(R(N) = e^{r(1-N/K)}\), we get: \[ N^* = e^{r(1-N^*/K)} N^* \] To find nontrivial fixed points, set \(N^*>0\), which simplifies to solve: \[ 1 = e^{r(1-N^*/K)} \]

Step 2: Solve for Nontrivial Fixed Points

From the equation \(1 = e^{r(1-N^*/K)}\), we take the natural log on both sides to yield: \[ 0 = r(1 - N^*/K) \] Simplifying, we find: \[ r - rN^*/K = 0 \] Solving for \(N^*\), we get: \[ rN^* = rK \] \[ N^* = K \] Therefore, the nontrivial fixed point is \(N^* = K\).

Step 3: Determine Stability by Analyzing Derivative

To determine stability, analyze the derivative of the function \(f(N) = R(N)N = e^{r(1-N/K)}N\) at \(N^*.\) The derivative is: \[ \frac{df}{dN} = \frac{d}{dN} \left(e^{r(1-N/K)}N \right) \] Using the product rule: \[ \frac{df}{dN} = e^{r(1-N/K)} + N \left(-\frac{r}{K}e^{r(1-N/K)}\right) \] Simplifying gives: \[ \frac{df}{dN} = e^{r(1-N/K)} (1 - \frac{rN}{K}) \] Evaluate this derivative at the fixed point \(N^*=K\): \[ \frac{df}{dN}\Big|_{N=K} = e^{r(1-K/K)} (1 - \frac{rK}{K}) = e^0 (1 - r) = 1 - r \] If \(1 - r < 1\), then the fixed point is stable since \(|1 - r| < 1\). Conversely, if \(r > 2\), the fixed point is unstable since \(|1 - r| > 1\).

Step 4: Conclude Stability Analysis

Based on the derivative analysis, the fixed point \(N^* = K\) is stable if \(0 < r < 2\). Once \(r > 2\), the fixed point becomes unstable. This is because, for stability, \(\left|1-r\right| < 1\) must hold true, which simplifies to \(-1 < 1-r < 1\).

Key Concepts

Nontrivial Fixed Points

In the study of population models, a fixed point is where the population size doesn't change over time. For our function, that means the population at time \(t+1\) is the same as at time \(t\). We call a fixed point 'nontrivial' if it's greater than zero, meaning we have an existing population.
To find a nontrivial fixed point in our model, we solve the equation:

• \(N^* = R(N^*) N^*\)

Substitute our given function for per capita growth rate, \(R(N) = e^{r(1-N/K)}\), to get:

• \(1 = e^{r(1-N^*/K)}\)

This equation must be solved for \(N^*\) greater than zero. After taking the natural logarithm and simplifying, we find that the only nontrivial fixed point is when the population equals the carrying capacity, \(N^* = K\). This means that population growth balances out precisely when the population size matches the environment's resources. Understanding how to find these points is crucial in modeling because they tell us where a population will stabilize in the absence of disturbances.

Stability Analysis

To determine how a population behaves around the nontrivial fixed point \(N^* = K\), we perform a stability analysis. This involves examining how small changes in population can affect its future growth. We do this by looking at the derivative of the function \(f(N) = R(N)N = e^{r(1-N/K)}N\). Take the derivative using the product rule:

• \(\frac{df}{dN} = e^{r(1-N/K)} (1 - \frac{rN}{K})\)

Then evaluate it at the fixed point \(N^* = K\):

• \(\frac{df}{dN}\Big|_{N=K} = 1 - r\)

The stability of the fixed point depends on \(|1 - r|\), where:- If \(|1 - r| < 1\), then disturbances will return to the fixed point, indicating stability. This occurs when \(0 < r < 2\).- If \(|1 - r| > 1\), disturbances will cause the population to move away from the fixed point, indicating instability. This happens if \(r > 2\).
Stability analysis is essential as it helps predict whether a population will maintain itself at a certain size or experience fluctuations.

Per Capita Growth Rate

The per capita growth rate, represented by \(R(N)\), is a critical component of population growth models. It describes how individual growth contributions add to the overall population size over time, adjusting for the current population density. In our model, the per capita growth rate function is given by

• \(R(N) = e^{r(1-N/K)}\)

This equation shows exponential growth but modified by the term \(1-N/K\), which accounts for the density-dependent effect. Here's how it works:- When \(N\) is well below \(K\), \(R(N)\) becomes larger, meaning the population has plenty of resources to grow quickly.- As \(N\) approaches \(K\), \(R(N)\) slows down, as resources become limited.- Once \(N\) equals \(K\), \(R(N) = 1\), meaning there's no per capita growth, leading to a stable population size.
This function reflects real-world scenarios where populations cannot grow indefinitely and must stabilize around a sustainable carrying capacity due to resource limits. Understanding \(R(N)\) provides insights into how environmental factors influence growth rates.