Question

In analyzing the Lotka-Volterra model of inter specific competition in Section \(14.2,\) three outcomes of competition between two species were identified. In three of the cases, one species out competes another, driving its population to zero. In the fourth case, the two species coexist. What condition is necessary for this outcome?

Short answer

Answer: The necessary condition for the coexistence of two species in the Lotka-Volterra competition model is when the competition coefficients, α and β, are smaller than the ratios of the carrying capacities of the respective species, i.e., α < (K1/K2) and β < (K2/K1). If this condition is met, both species can coexist in stable populations.

Step-by-step solution

Understanding the Lotka-Volterra competition model

The Lotka-Volterra competition model is a set of general equations used to model interspecific competition between two species. The model is represented by the following system of ordinary differential equations (ODEs): \[\frac{dN_1}{dt} = r_1 N_1 \left(1 - \frac{N_1 + \alpha N_2}{K_1}\right)\] \[\frac{dN_2}{dt} = r_2 N_2 \left(1 - \frac{N_2 + \beta N_1}{K_2}\right)\] where \(N_1\) and \(N_2\) represent the population sizes of species 1 and 2, \(r_1\) and \(r_2\) are their intrinsic rates of population growth, \(K_1\) and \(K_2\) are the carrying capacities of the environment for each species, and \(\alpha\) and \(\beta\) are the competition coefficients.

Identify the equilibrium points

To identify the conditions that lead to coexistence, we need to find the equilibrium points for the system of ODEs. An equilibrium point is a point where the populations of both species remain stable, i.e., the population sizes don't change over time. Mathematically, this means that \(\frac{dN_1}{dt} = 0\) and \(\frac{dN_2}{dt} = 0\). By setting \(\frac{dN_1}{dt} = 0\) and \(\frac{dN_2}{dt} = 0\), we get the equilibrium points: \[\begin{cases} Case 1: N_1 = 0,~ N_2 = 0 \newline Case 2: N_1 = K_1,~ N_2 = 0 \newline Case 3: N_1 = 0,~ N_2 = K_2 \newline Case 4: N_1 = K_1 - \alpha K_2,~ N_2 = K_2 - \beta K_1 \end{cases}\]

Determine the condition for coexistence

The Case 4 equilibrium point represents the coexistence of species 1 and 2, where both populations are stable and nonzero. However, there is a crucial condition for this outcome. The condition necessary for coexistence is that both species' population sizes are positive and within their respective carrying capacities, i.e., \(N_1 > 0\) and \(N_2 > 0\) and \(N_1 < K_1\) and \(N_2 < K_2\). From Case 4, we have: \[N_1 = K_1 - \alpha K_2 > 0\] \[N_2 = K_2 - \beta K_1 > 0\] For these inequalities to hold, we require the following condition: \[\alpha < \frac{K_1}{K_2} \,\text{ and}\, \beta < \frac{K_2}{K_1}\]

Summarize the condition for coexistence

In conclusion, the condition necessary for the coexistence of two species in the Lotka-Volterra competition model is: \[\alpha < \frac{K_1}{K_2} \,\text{ and}\, \beta < \frac{K_2}{K_1}\] This means that the competition coefficients, \(\alpha\) and \(\beta\), must be smaller than the ratios of the carrying capacities of the respective species. If this condition is met, both species can coexist in stable populations.

Key concepts

Interspecific Competition

In the study of ecology, interspecific competition refers to the struggle between different species as they compete for the same resources within an ecosystem. This can include food, space, water, or even light for photosynthetic organisms. Interspecific competition can have a significant impact on the populations of the species involved, shaping community structures, and influencing biodiversity. The Lotka-Volterra competition model is a mathematical representation of this biological concept, allowing us to predict the outcomes of competition between two species.

For students aiming to understand this model, it's important to note that these outcomes can range from one species being competitively excluded (outcompeted to extinction) to an equilibrium where both species coexist. The Lotka-Volterra model simplifies the complexity of ecosystems into equations, which describe how the population of each species grows and is affected by the presence of the other. The coefficients \(\alpha\) and \(\beta\) within the equations quantify the extent of competitive impact one species has on another. Greater values of these coefficients indicate stronger competition. The competition model thus gives us valuable insight into how different species might share or compete for their ecological niches.

Equilibrium Points

In the context of the Lotka-Volterra competition model, equilibrium points are conditions where the populations of competing species reach a state of balance—neither population grows or shrinks. These points are critical in understanding the long-term dynamics of species interactions. By setting the rate of change in populations to zero, we find various scenarios, including the potential for stable coexistence.

For educators and students alike, equilibrium points are pivotal to grasping how species can influence one another's presence within an ecosystem. This can be visualized as points on a graph where the population sizes level off. The fourth case from the model, where both species coexist, is one where both populations stabilize at a non-zero level. This indicates that species can share a habitat without driving each other to extinction, should the competition coefficients allow it. However, this equilibrium is delicate, as changes in environmental factors or the introduction of new competitors can disrupt the balance.

Carrying Capacity

The carrying capacity, denoted by \(K\) in the Lotka-Volterra model, defines the maximum population size of a species that its environment can sustain indefinitely. Carrying capacity is determined by the availability of resources, such as food and habitat space, and by the effect of predators or diseases.

This concept is vital for both the predator-prey model and the competition model. In the competitive context, each species has its own carrying capacity which can be affected by the presence of competing species. The condition for stable coexistence in the competition model requires that the competition coefficients are low enough so that the populations of both species are below their respective carrying capacities. If the coefficients are too high, one species overshadows the other, leading to competitive exclusion. Students should consider carrying capacities as dynamic ceilings for population growth, influenced by interspecies relationships and environmental constraints.