Problem 22 - Calculus for Biology and Medicine

Question

A commonly used model for the density-dependent dynamics of a population is the recurrence equation: $$N_{t+1}=R N_{t}^{b}$$ where \(b, R>0\) are constants that take different values depending on the species of organism that is being modeled and its habitat. When \(b=1\), the model predicts that the population will grow exponentially. (a) Show that the population has a non-trivial equilibrium point (that is \(N \neq 0\) ), that you should determine. (b) Show that if \(-11\) ? Let \(b=2\) and \(R=1\), so \(N_{t+1}=N_{t}^{2}\). Find all of the equilibria of the recursion relation, and determine which (if any) are stable. (d) Calculate the first ten terms of the recursion relation \(N_{t+1}=\) \(N_{t}^{2}\) if (i) \(N_{0}=0.5\) and (ii) \(N_{0}=2 ?\) (e) If \(N_{t+1}=N_{t}^{2}\), what are the possible behaviors of the population as \(t \rightarrow \infty\) ?

Step-by-Step Solution

Answer

Non-trivial equilibrium is \(N = \left( \frac{1}{R} \right)^{\frac{1}{1-b}}\). Stable for \(-1 1\).

Step 1: Find the Equilibrium Point

To find the equilibrium point, set \(N_{t+1} = N_t\). This gives the equation: \[ R N_{t}^{b} = N_t \]Rearranging gives:\[ R N_{t}^{b} = N_t \Rightarrow R = \frac{N_t}{N_t^b} = N_t^{1-b} \]For there to be a non-trivial equilibrium (\(Neq 0\)), solve:\[ N^{1-b} = \frac{1}{R} \]The non-trivial equilibrium point is \(N = \left( \frac{1}{R} \right)^{\frac{1}{1-b}}\) if \(b eq 1\).

Step 2: Stability of Equilibrium for -1 < b < 1

To check for stability, compute the derivative of the recurrence relation at equilibrium:\[ N_{t+1} = R N_{t}^{b} \Rightarrow f(N) = R N^b \]The derivative is: \[ f'(N) = R b N^{b-1} \]Substitute the equilibrium calculated: \[ f'(\bar{N}) = R b \left( \frac{1}{R} \right)^{\frac{b-1}{1-b}} = b \left( \frac{1}{R} \right)^{\frac{b-1}{1-b}} \]For the equilibrium to be stable, \(|f'(\bar{N})| < 1\). Given \(-1 < b < 1\), the expression simplifies and shows that the condition \(|b| < 1\) ensures stability for any \(R\).

Step 3: What Happens When b > 1: Example b = 2

If \(b > 1\), the behavior changes. Consider \(b=2\) and \(R=1\), with the recurrence:\[ N_{t+1} = N_t^2 \]Equilibrium points occur at \(N_{t+1} = N_t\), leading to:\[ N_t = N_t^2 \Rightarrow N_t(N_t - 1) = 0 \]Solutions are either \(N_t = 0\) or \(N_t = 1\). Check stability by evaluating the derivative at these points:\[ f'(N) = 2N \]- At \(N=0\), \(f'(0) = 0\) which indicates potential stability.- At \(N=1\), \(f'(1) = 2\), implying instability as \(|2| > 1\).

Step 4: Calculate First Ten Terms for N_0 = 0.5 and N_0 = 2

(i) Start with \(N_0 = 0.5\):Calculate iteratively using \(N_{t+1} = N_t^2\):1. \(N_1 = 0.5^2 = 0.25\)2. \(N_2 = 0.25^2 = 0.0625\)3. \(N_3 = 0.0625^2 = 0.00390625\)Continue similarly to find:- \(N_4 = 1.52588 \times 10^{-5}\)- \(N_5 = 2.32831 \times 10^{-10}\)- Terms continue diminishing towards 0.(ii) Start with \(N_0 = 2\):Calculate:1. \(N_1 = 2^2 = 4\)2. \(N_2 = 4^2 = 16\)3. \(N_3 = 16^2 = 256\)Subsequent terms grow very large rapidly.

Step 5: Behavior as t → ∞ if N_{t+1} = N_t^2

If \(N_{t} < 1\), successive iterations decrease and \(N_t\) approaches 0 (the series converges). If \(N_t > 1\), the series diverges to infinity unless an equilibrium is reached.

Key Concepts

Density-Dependent Model

The density-dependent model in population dynamics is a mathematical framework that describes how populations change over time, taking into consideration the effects of population density. As a population grows, various factors such as competition for resources, predation, disease, and territorial behavior tend to increase, affecting birth and death rates. The model's general formula is given by the recurrence equation:

• \[ N_{t+1} = R N_t^b \]
• where \(N_t\) is the population at time \(t\), \(R\) is a growth rate constant, and \(b\) is a constant that influences the nature of the growth.
• \(R\) and \(b\) vary based on species and environmental factors.
• When \(b = 1\), the model predicts exponential growth.

This model is termed 'density-dependent' because the rate of population growth depends on the current population size, reflected by the parameter \(b\). This relationship allows the model to characterize how a population's growth can slow as it reaches the carrying capacity of its environment.

Recurrence Equation

The recurrence equation is a fundamental tool used to predict future population sizes based on current data. It essentially provides a formula or rule whereby the population size at one time step informs the population size at the next time step.

• In our context, the recurrence equation is: \[ N_{t+1} = R N_t^b \]
• This equation helps to iteratively calculate the size of the population across successive time periods.
• Unlike simple growth models, it takes into account any changes in growth rate due to fluctuations in population size.

The recurrence equation provides a recursive way to project population dynamics, allowing for detailed examination of population changes over time. It adjusts the population size prediction by considering specific growth rates and patterns dictated by the constants \(R\) and \(b\). This feature makes it a versatile tool in ecological modeling.

Equilibrium Stability

Equilibrium stability in population dynamics refers to the conditions under which the population size of a species stabilizes over time. An equilibrium point occurs when the population size stops changing, i.e., \(N_{t+1} = N_t\). In the density-dependent model, the non-trivial equilibrium is achieved when the population reaches a point where that balance is perfect:

• This can be found by setting \( f(N) = N \): \[ R N_t^b = N_t \]
• Thus, for non-trivial equilibrium (i.e., \(N eq 0\)), solve: \[ N^{1-b} = \frac{1}{R} \]

The stability of this equilibrium depends on the derivative of the function. Under the condition \(-1 < b < 1\), the system is stable because any small deviation in population size willtend to return to equilibrium. Stability is critically evaluated by ensuring the condition \(|f'(\bar{N})| < 1\), where \(\bar{N}\) is the equilibrium point.

Exponential Growth Model

The exponential growth model is a special case of the density-dependent model. It represents a scenario where the population grows proportionally to its current size without complicating factors of limited resources or other density effects.

• In mathematical terms, for the recurrence equation to reflect exponential growth, we need \(b = 1\), leading to:
• \[ N_{t+1} = R N_t \]
• This simple form implies a steady growth rate that does not diminish regardless of the population size.

While exponential growth describes rapid increase, it is not sustainable over the long term due to ecological limits. It is particularly useful for understanding short-term dynamics or initial growth phases, allowing students to see the contrast between unrestricted and density-dependent growth. The exponential model is characterized by a constant doubling time, highlighting its runaway nature unless tempered by environmental or resource constraints.