Problem 11 - 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 coexist: Species 1 reaches an equilibrium of about 180 and species 2 reaches an equilibrium of about \(80 .\) If their densities follow the LotkaVolterra equation of interspecific competition, find \(\alpha_{12}\) and \(\alpha_{21}\).

Step-by-Step Solution

Answer

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

Step 1: Understand the Lotka-Volterra Equations

The Lotka-Volterra equations for two species in competition are given by:\[\frac{dN_1}{dt} = r_1 N_1 \left(1 - \frac{N_1 + \alpha_{12} N_2}{K_1}\right)\]\[\frac{dN_2}{dt} = r_2 N_2 \left(1 - \frac{N_2 + \alpha_{21} N_1}{K_2}\right)\]where \(N_1\) 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 Equations for Equilibrium

At equilibrium, the growth rate is zero for both species:\[0 = r_1 N_1 \left(1 - \frac{N_1 + \alpha_{12} N_2}{K_1}\right)\]\[0 = r_2 N_2 \left(1 - \frac{N_2 + \alpha_{21} N_1}{K_2}\right)\]

Step 3: Plug in Equilibrium Values

For species 1 alone: \(N_1 = 200, N_2 = 0\). The carrying capacity \(K_1 = 200\). For species 2 alone: \(N_2 = 150, N_1 = 0\). The carrying capacity \(K_2 = 150\).With both together: \(N_1 = 180, N_2 = 80\).

Step 4: Solve for \(\alpha_{12}\)

Substitute \(N_1 = 180\), \(N_2 = 80\) into the equation:\[0 = 180 \left(1 - \frac{180 + \alpha_{12} \cdot 80}{200}\right)\]Solving gives:\[\alpha_{12} = \frac{200 - 180}{80} = \frac{20}{80} = 0.25\]

Step 5: Solve for \(\alpha_{21}\)

Substitute \(N_2 = 80\), \(N_1 = 180\) into the equation:\[0 = 80 \left(1 - \frac{80 + \alpha_{21} \cdot 180}{150}\right)\]Solving gives:\[\alpha_{21} = \frac{150 - 80}{180} = \frac{70}{180} \approx 0.39\]

Step 6: Conclusion: Values of Competition Coefficients

The competition coefficients \(\alpha_{12}\) and \(\alpha_{21}\) are found to be 0.25 and 0.39 respectively.

Key Concepts

Interspecific Competition

Interspecific competition occurs when two or more species compete for the same limited resources, such as food, space, or light. This type of competition can significantly impact population dynamics and biodiversity within an ecosystem. When species compete, they influence each other's growth rates, reproduction, and survival. Over time, interspecific competition can lead to the adaptation of species, enabling them to coexist through resource partitioning or other mechanisms.
Understanding interspecific competition is crucial because it helps explain how species coexist in nature, even when they rely on similar resources. In our scenario with the two beetle species, the interspecific competition affects their population sizes when they are together, rather than when they are reared separately.
The Lotka-Volterra equations are often used to model this competition mathematically. This example demonstrates that even when competing, species may reach a new equilibrium, indicating that they've adjusted their resource utilization or faced some trade-offs to survive together.

Competition Coefficients

Competition coefficients, denoted as \(\alpha_{12}\) and \(\alpha_{21}\) in the Lotka-Volterra model, help quantify the impact one species has on another. Specifically, \(\alpha_{12}\) measures the impact of species 2 on the growth rate of species 1, while \(\alpha_{21}\) assesses the effect of species 1 on species 2.
These coefficients provide insight into how strongly one species competes with another for resources. For instance, an \(\alpha_{12}\) value of 0.25 implies that each individual of species 2 has one-quarter the effect on species 1's resources as an individual of species 1. Similarly, \(\alpha_{21} \approx 0.39\) indicates that species 1 individuals have a noticeable impact on species 2's resources.
In practical terms, knowing these coefficients allows ecologists to predict which species may dominate, coexist, or become extinct in shared habitats over time. It's a powerful tool for understanding competitive interactions and guiding conservation efforts.

Equilibrium Population

The concept of an equilibrium population refers to a stable state where the population size remains constant over time, assuming no changes in external conditions. In a biological sense, it means that the birth rate equals the death rate, leading to a stable number of individuals.
In the case of the beetles, equilibrium populations are reached both when the species are isolated and when they coexist. Alone, species 1 and 2 reach equilibrium populations of 200 and 150, respectively. Together, they adjust to 180 and 80, thanks to interspecific competition.
Reaching an equilibrium population is essential for species survival. It indicates that the population has adapted to the current environmental conditions and resource availability. However, disturbances or changes in the environment can shift this balance, leading to population growth or decline.

Carrying Capacity

Carrying capacity is a critical concept in ecology that refers to the maximum population size of a species that an environment can sustain indefinitely, given the available resources such as food, habitat, water, and other necessities.
Each species has its carrying capacity, which can vary depending on environmental conditions and resource availability. In our beetle example, species 1 has a carrying capacity of 200 when alone, and species 2's carrying capacity is 150.
However, when resources become more limited due to interspecific competition, both species adjust their populations below their initial carrying capacities. This adjustment highlights how competition and resource limitation interact to influence population sizes.
Understanding carrying capacity is vital for managing wildlife populations and ecosystems, as it helps in predicting how species will respond to environmental changes and resource management practices.