Problem 12 - Calculus for Biology and Medicine

Question

Suppose that two species of beetles are reared together. Species 1 wins if there are initially 100 individuals of species 1 and 20 individuals of species \(2 .\) But species 2 wins if there are initially 20 individuals of species 1 and 100 individuals of species \(2 .\) When the beetles are reared separately, both species seem to reach an equilibrium of about \(120 .\) On the basis of this information and assuming that the densities follow the Lotka-Volterra model of interspecific competition, can you give lower bounds on \(\alpha_{12}\) and \(\alpha_{21} ?\)

Step-by-Step Solution

Answer

Lower bounds are \(\alpha_{12} > \frac{5}{6}\) and \(\alpha_{21} > \frac{1}{6}\).

Step 1: Understanding the Lotka-Volterra model

The Lotka-Volterra model of interspecific competition is a set of differential equations used to model the population dynamics of two species competing for the same resources. It includes competition coefficients \(\alpha_{12}\) and \(\alpha_{21}\), which represent the impact of species 2 on species 1 and the impact of species 1 on species 2, respectively.

Step 2: Equilibrium analysis when species are separate

When both species are reared separately, each reaches an equilibrium population of 120 individuals. This suggests their growth without interspecies competition leads to self-limiting behavior, where the carrying capacity \(K\) is approximately 120.

Step 3: Initial condition analysis for dominance

Species 1 wins when 100 individuals of species 1 face 20 individuals of species 2 indicating \(\alpha_{12} < \frac{100}{120} = \frac{5}{6}\) as a minimum effect species 2 can have on species 1 to not win. Conversely, species 2 wins when 20 individuals of species 1 face 100 individuals of species 2 suggesting \(\alpha_{21} < \frac{20}{120} = \frac{1}{6}\) as a minimum effect species 1 can have on species 2 to not win.

Step 4: Conclusion and determination of lower bounds

To ensure species 2 does not win the initial condition 1 and species 1 does not win initial condition 2, we conclude that the competition coefficients must satisfy \(\alpha_{12} > \frac{5}{6}\) and \(\alpha_{21} > \frac{1}{6}\). These reflect that species 2 must have a significant effect on species 1 in a competitive scenario and vice versa.

Key Concepts

Interspecific Competition

Interspecific competition occurs when individuals of different species compete for the same resources in an environment. This can include food, space, or other essentials they both rely on for survival. In the Lotka-Volterra model, this competition is represented mathematically. Each species affects the growth and survival of the other, influencing their overall population dynamics.

Let's imagine two species of beetles that both need the same food source. If there is food shortage, one species might outcompete the other, leading to the dominated species' decline. However, if both have similar competitive strengths, they might reach a balance—a state where neither species goes extinct, but both remain stably present. Understanding how they interact with each other helps us learn more about biodiversity and ecological balance in natural habitats.

Competition Coefficients

Competition coefficients in the Lotka-Volterra model quantify the impact that one species has on another's growth. The coefficients, typically written as \(\alpha_{12}\) and \(\alpha_{21}\), are essential in predicting which species will dominate when they are both present.

Here's how it works:

• \(\alpha_{12}\): This coefficient measures the impact of species 2 on species 1. A higher value means species 2 severely affects species 1's ability to grow.
• \(\alpha_{21}\): This coefficient measures the impact of species 1 on species 2. Similarly, a high \(\alpha_{21}\) implies a significant effect on species 2 by species 1.

In our beetle scenario, if we find \(\alpha_{12} > \frac{5}{6}\), it indicates that species 2 has more than enough competitive pressure on species 1, preventing species 1 from thriving when they are competing directly. Conversely, \(\alpha_{21} > \frac{1}{6}\) means species 1 significantly impacts the survival of species 2, pressuring species 2 when in competition.

Population Dynamics

Population dynamics involve the study of how and why the number of individuals in a species changes over time. When analyzing populations that interact, like in the case of our beetles, models like Lotka-Volterra become invaluable. They offer insights into the factors that control population size and growth.

Population dynamics consider many elements, such as:

• Birth rates and death rates: These are fundamental in understanding how populations grow or shrink.
• Immigration and emigration: They influence how populations can expand or contract based on individuals moving in or out of a population.
• Carrying capacity: This is the maximum population size an environment can sustain indefinitely.

For beetles reared in isolation, reaching an equilibrium of 120 individuals suggests they hit their carrying capacity. In a competitive scenario, interactions might lower this equilibrium, affecting both species' survival and growth. Thus, population dynamics show us how environmental and interspecies interactions drive changes in species numbers.