Problem 9 - Calculus for Biology and Medicine

Question

Suppose that two species of beetles are reared together in one experiment and separately in another. When species 1 is reared alone, it reaches an equilibrium of about \(200 .\) When species 2 is reared alone, it reaches an equilibrium of about \(150 .\) When both of them are reared together, they seem to be able to \(\mathrm{co}^{-}\) exist: Species 1 reaches an equilibrium of about 180 and species 2 reaches an equilibrium of about \(80 .\) If their densities follow the Lotka-Volterra equation of interspecific competition, find \(\alpha_{12}\) and \(\alpha_{21}\).

Step-by-Step Solution

Answer

\(\alpha_{12} = 0.25\) and \(\alpha_{21} \approx 0.39\).

Step 1: Understand the Lotka-Volterra Equations

The Lotka-Volterra equations for interspecific competition describe how the populations of two species influence each other. The equations are: \[\begin{align*}\frac{dN_1}{dt} &= r_1N_1\left(1 - \frac{N_1 + \alpha_{12}N_2}{K_1}\right), \\frac{dN_2}{dt} &= r_2N_2\left(1 - \frac{N_2 + \alpha_{21}N_1}{K_2}\right)\end{align*}\]where \(r_1\) and \(r_2\) are the intrinsic growth rates, \(N_1\) and \(N_2\) are the population sizes, \(K_1\) and \(K_2\) are the carrying capacities, and \(\alpha_{12}\) and \(\alpha_{21}\) are the competition coefficients.

Step 2: Set Up Equations for Species 1 Alone

When species 1 is reared alone, its population reaches an equilibrium of \(200\), which is its carrying capacity \(K_1 = 200\). Therefore, the equation for species 1 when reared alone simplifies to: \[ N_1 = 200 \].

Step 3: Set Up Equations for Species 2 Alone

Similarly, when species 2 is isolated, it reaches a carrying capacity of \(150\), so \(K_2 = 150\) by the equation \( N_2 = 150 \).

Step 4: Analyze Joint Rearing Equilibrium

When reared together, species 1 and 2 reach equilibriums of \(180\) and \(80\) respectively. Substituting these values into the equations gives: \[ 180 = 200 - \alpha_{12} \times 80 \] and \[ 80 = 150 - \alpha_{21} \times 180 \].

Step 5: Solve for \(\alpha_{12}\)

Rearrange the equation for species 1's joint equilibrium: \[ \alpha_{12} \times 80 = 200 - 180 \Rightarrow \alpha_{12} \times 80 = 20 \]. Solving for \(\alpha_{12}\) yields \( \alpha_{12} = \frac{20}{80} = 0.25 \).

Step 6: Solve for \(\alpha_{21}\)

Rearrange the equation for species 2's joint equilibrium: \[ \alpha_{21} \times 180 = 150 - 80 \Rightarrow \alpha_{21} \times 180 = 70 \]. Solving for \(\alpha_{21}\) gives \( \alpha_{21} = \frac{70}{180} \approx 0.39 \).

Key Concepts

Interspecific Competition

Interspecific competition is a fascinating and complex concept in ecology. It occurs when individuals of different species compete for the same resources in an ecosystem, such as food, space, or light. This type of competition often influences the population sizes of the species involved. In the case of the beetles in our exercise, both species are competing for the same resources when reared together. The Lotka-Volterra equations, which are used to describe this competition, provide us with a framework to understand how two different species affect each other's population growth. The equations include competition coefficients (\( \alpha_{12} \) and \( \alpha_{21} \)), which represent the impact one species has on the other's growth rate relative to its own. A higher coefficient means more significant impact. By calculating these coefficients, we can predict how species will interact and coexist over time, providing insight into ecosystem dynamics. In our beetle example, the existence of a competition coefficient ratio of 0.25 and 0.39 reveals a moderate competition level, allowing both species to coexist but at reduced population sizes compared to their isolated environments.

Carrying Capacity

Carrying capacity refers to the maximum population size of a species that an environment can sustain indefinitely. This concept is crucial in population dynamics as it determines the limits to the growth of that species. In our exercise scenario, when the species were raised separately, species 1 had a carrying capacity of 200, while species 2 had a carrying capacity of 150. This means that when no interspecific competition is present, each species can grow up to these population limits, given the available resources and environmental conditions. Carrying capacity is influenced by several factors, such as resource availability, environmental conditions, and species-specific requirements. Understanding this concept helps us comprehend the balance ecosystems maintain, as populations adjust based on the carrying capacity. When both species are together, their carrying capacities are reduced due to competition, illustrating that resources are shared and can limit growth.

Population Dynamics

Population dynamics is the study of how and why populations change over time, a key element in understanding ecological balance and the interactions within ecosystems. It encompasses factors such as birth rates, death rates, immigration, emigration, and competition like we see with the Lotka-Volterra equation. These dynamics help predict future changes in populations and potential outcomes for ecosystems. In the exercise, population dynamics is showcased through the changes in the beetle populations as they interact. Alone, each species reaches its respective carrying capacity, but together, interspecific competition modifies their growth dynamics. The Lotka-Volterra model illustrates how population dynamics result from complex interactions between two competing species. These interactions may lead to equilibrium where both species coexist. Alternatively, one may outcompete the other, leading to a significant population decline or extinction of the less competitive species. Understanding these dynamics aids conservation efforts by predicting how species populations might respond to environmental changes or human impacts.