Problem 23 - Calculus for Biology and Medicine

Question

The Ricker model was introduced by Ricker (1954) as an alternative to the discrete logistic equation to describe the density-dependent growth of a population. Under the Ricker model the population \(N_{t}\) sampled at discrete times \(t=0,1,2, \ldots\) is modeled by a recurrence equation $$N_{t+1}=R_{0} N_{t} \exp \left(-a N_{t}\right)$$ where \(R_{0}\) and \(a\) are positive constants that will vary between different species and between different habitats. (a) Explain why you would expect \(R_{0}>1\) (Hint: consider the population growth when \(N_{t}\) is very small.) (b) Show that the recursion relation has two equilibria, a trivial equilibrium (that is, \(N=0\) ) and another equilibrium, which you should find. (c) Show that if \(R_{0}>1\) then use the stability criterion for equilibria to show that the trivial equilibrium point is unstable. (d) Use the stability criterion for equilibria to show that the nontrivial equilibrium point is stable if \(0<\ln R_{0}<2\). (e) If \(R_{0}>1\) then \(\ln R_{0}>0\), so most populations will meet the first inequality condition. What happens if \(\ln R_{0}>2 ?\) Let's try some explicit values: \(R_{0}=10, a=1, N_{0}=1 .\) Calculate the first ten terms of the sequence, and describe in words how the sequence behaves.

Step-by-Step Solution

Answer

When \(R_0 > 1\), small populations grow. Two equilibria exist: \(N = 0\) (unstable for \(R_0 > 1\)) and \(N = \frac{\ln R_0}{a}\) (stable if \(0 < \ln R_0 < 2\)). For \(\ln R_0 > 2\), population dynamics become chaotic.

Step 1: Understanding the R_0 > 1 Condition

We expect \(R_0 > 1\) because the population should grow when it is very small (close to zero). If \(R_0 \leq 1\), the term \(R_0 N_t\) would lead to a population decrease or a stable state rather than growth.

Step 2: Finding Equilibrium Points

An equilibrium occurs when the population size remains constant, therefore setting \(N_{t+1} = N_t = N\) gives the equation \(N = R_0 N \exp(-aN)\). Either \(N = 0\), which is the trivial equilibrium, or solving \(1 = R_0 \exp(-aN)\) leads to the non-trivial equilibrium: \(N = \frac{\ln R_0}{a}\).

Step 3: Analyzing Stability of the Trivial Equilibrium

To check stability, calculate the derivative of the function \(f(N) = R_0 N \exp(-a N)\) at the equilibrium \(N = 0\). We find \(f'(N) = R_0 (1 - aN) \exp(-aN)\). At \(N = 0\), this simplifies to \(f'(0) = R_0\). Since \(R_0 > 1\), the derivative is greater than 1, indicating that \(N = 0\) is unstable.

Step 4: Establishing Stability of Non-Trivial Equilibrium

The non-trivial equilibrium \(N = \frac{\ln R_0}{a}\) is evaluated by substituting back into \(f(N)\). The stability criterion requires \(|f'(N)| < 1\). Computing \(f'(N)\), and substituting the non-trivial equilibrium gives \(f'\left(\frac{\ln R_0}{a}\right) = 1 - \ln R_0\). Stability requires \(0 < \ln R_0 < 2\).

Step 5: Observing Behavior when \(\ln R_0 > 2\)

Using \(R_0 = 10, a = 1, N_0 = 1\), we compute successive terms: \(N_1 = 10 \cdot 1 \cdot \exp(-1)\), \(N_2 = 10 \cdot N_1 \cdot \exp(-N_1)\), continuing for 10 terms. The sequence behaves chaotically due to the large \(\ln R_0\), diverging from stability.

Key Concepts

Discrete Logistic Equation

The discrete logistic equation is a mathematical representation used to model the dynamics of populations over time. It is particularly useful for understanding how populations grow and stabilize in environments with limited resources. The equation takes the form of a recurrence relation, where population size at the next time step depends on the current size and some growth rate parameter. This equation helps in:

• Predicting future population sizes.
• Exploring how populations react to various growth constraints.

It's often used in ecology and biology to predict behaviors of real populations in discrete time scenarios, like yearly counts of animals or plants. By using this model, one can visualize how population size fluctuates between reproduction and resource availability limits.

Population Growth Models

Population growth models are crucial in ecological and biological studies as they help analyze and predict the growth patterns of populations. These models like the Ricker Model illustrate how populations change in size over time due to births, deaths, and immigration. They provide insights into different aspects:

• The potential for populations to expand given optimal conditions.
• How populations reach a balance when faced with environmental limits.
• The impact of different species and habitats on population growth rates.

These models can be continuous or discrete, enabling researchers to simulate different real-world contexts and observe how populations respond to them effectively.

Equilibrium Stability

Understanding equilibrium stability in population models helps assess whether a population will settle into a balanced state over time. An equilibrium is stable if, after small disturbances, the population returns to that state. In models like the Ricker model, this involves:

• Calculating points where the population does not change from one time step to the next.
• Using derivatives to determine whether these points are stable or unstable.
• Distinguishing between trivial (e.g., population extinction) and non-trivial (e.g., steady-state population size) equilibria.

An equilibrium is considered stable if conditions of growth rates and population sizes ensure return to equilibrium after small changes. This concept helps in understanding long-term population sustainability.

Density-Dependent Growth

Density-dependent growth is a concept where population growth rate changes based on the population size itself. As the population increases, the available resources per individual decrease, leading to reduced growth rates. In the context of the Ricker model,:

• The growth rate is affected by population density due to factors like limited food supply, space, and other resources.
• The model incorporates density dependence through an exponential decay function that reduces growth as population size increases, combining both growth potential and resource limitations to predict population sizes over time.

Understanding density-dependent growth is essential for conservation efforts and resource management strategies as it reflects the reality of natural environmental constraints on population growth.