Problem 7 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Consider a modification of the Lotka-Volterra equations for competition between two species in which \(\alpha_{1,2}=0\). $$\begin{array}{l}p_{1}^{\prime}(t)=r_{1} \times p_{1}(t) \times\left(1-\frac{p_{1}(t)+0 \times p_{2}(t)}{M_{1}}\right)=r_{1} \times p_{1}(t) \times\left(1-\frac{p_{1}(t)}{M_{1}}\right) \\ p_{2}^{\prime}(t)=r_{2} \times p_{2}(t) \times\left(1-\frac{p_{2}(t)+\alpha_{2,1} p_{1}(t)}{M_{2}}\right) \end{array}$$ Thus population 1 is not affected by population 2 but population 2 is affected by population \(1 .\) Suppose that $$00\) then \(z(t) \rightarrow 0\) as \(t \rightarrow \infty\) c. What happens to the second population if \(M_{2}>\alpha_{2,1} M_{1} ?\)

Step-by-Step Solution

Answer

1. Population 1 stabilizes at \( M_1 \). 2. If \( M_2 < \alpha_{2,1} M_1 \), population 2 goes to zero. 3. If \( M_2 > \alpha_{2,1} M_1 \), population 2 can stabilize at a positive level.

Step 1: Analyze the First Population

Consider the equation for the first population, \( p_1(t) \):\[ p_1^{\prime}(t) = r_1 \times p_1(t) \times\left(1-\frac{p_1(t)}{M_1}\right) \]This represents the logistic growth model. As \( t \to \infty \), if \( 0 < p_1(0) < M_1 \), \( p_1(t) \) will approach the equilibrium \( M_1 \), since the term \( 1-\frac{p_1(t)}{M_1} \) becomes 0 when \( p_1(t) = M_1 \). Thus, \( p_1(t) \rightarrow M_1 \).

Step 2: Analyze the Second Population with M2 < α2,1 M1

Consider the equation for the second population, \( p_2(t) \):\[ p_2^{\prime}(t) = r_2 \times p_2(t) \times\left(1-\frac{p_2(t)+\alpha_{2,1} p_1(t)}{M_2}\right) \]We assume \( p_1(t) = M_1 \) for \( t \geq t_0 \). If \( M_2 < \alpha_{2,1} M_1 \), the condition \( -k z(t) \) is fulfilled, i.e., \( p_2^{\prime}(t) < -k p_2(t) \), leading to exponential decay. Therefore, \( p_2(t) \rightarrow 0 \) as \( t \rightarrow \infty \).

Step 3: Analyze the Second Population with M2 > α2,1 M1

If \( M_2 > \alpha_{2,1} M_1 \), the second population has a chance to sustain itself or potentially grow. Specifically:1. Substitute \( p_1(t) = M_1 \) in the equation for \( p_2(t) \):\[ p_2^{\prime}(t) = r_2 \times p_2(t) \times\left(1-\frac{p_2(t)+\alpha_{2,1} M_1}{M_2}\right) \] 2. This results in a positive growth factor because:\[ 1-\frac{\alpha_{2,1} M_1}{M_2} > 0 \]This suggests that \( p_2(t) \) can remain positive or increase, thus potentially stabilizing at a positive value.

Key Concepts

population dynamics

Population dynamics is crucial to understanding how populations of different species change over time. It involves analyzing various factors that affect the growth and decline of populations. The Lotka-Volterra model is a foundational concept in this field, especially when discussing competitive interactions among species.
In this particular case, we observe how the dynamic interaction between two species influences their population sizes.

• For the first population, the equation depends solely on its own size and resources available from the environment.
• The second population, however, is also affected by the presence of the first population, introducing a level of complexity.

Understanding these dynamics requires examining how growth rates, carrying capacities, and interaction coefficients play a role in species competition.
Each species aims to maximize its population but must navigate the limitations imposed by other species and the environment.

competitive species interaction

Competitive species interaction explains how different species compete for limited resources such as food, space, and light. The Lotka-Volterra model simplifies these interactions by quantifying how species affect each other's growth rates.
In the model given,

• the first species does not affect the second one (\(\alpha_{1,2} = 0\)), meaning the second species faces less competition than it might otherwise.
• Meanwhile, the second species is affected by competition with the first (\(\alpha_{2,1}\)).

This situation mirrors real ecological phenomena where species might have different degrees of competitive pressure.
When \(M_{2} < \alpha_{2,1} M_{1}\), the competitive pressure is so intense that the second population cannot sustain itself and declines to zero.
If instead \(M_{2} > \alpha_{2,1} M_{1}\), it suggests that the second population can tolerate or circumvent the competitive pressure, potentially allowing both species to coexist, albeit with persistent competition.

logistic growth model

The logistic growth model is essential for understanding how populations grow in environments with limited resources. It is characterized by a growth rate that is initially exponential, but slows as the population reaches the carrying capacity of the environment.
Mathematically, this is expressed as:\[ p^{\prime}(t) = r \times p(t) \times\left(1-\frac{p(t)}{M}\right) \]where:

• \(p(t)\)is the population size at time \(t\),
• \(r\)is the intrinsic growth rate, and
• \(M\)is the maximum carrying capacity.

In our scenario, the first species follows this model independently of the second species due to \(\alpha_{1,2} = 0\).
The species grows until it reaches its carrying capacity \(M_{1}\). For the second species, the growth depends not only on its own population size but also on the impact of the first species, which is a deviation from the classic model, inserting a real-world complexity into the theoretical framework.