Question

In analyzing the Lotka-Volterra model of interspecific competition in Section \(13.3,\) four outcomes of competition between two species were identified. In three of the cases, one species outcompetes 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 coexistence in the Lotka-Volterra competition model is that the competition coefficients must be less than the intrinsic growth/death rates of the respective species: \(b_x < a_x\) and \(d_y < c_y\). This condition ensures that the population growth of one species is not suppressed too much by the presence of the other species.

Step-by-step solution

Understand the Lotka-Volterra competition model

The Lotka-Volterra model of interspecific competition is described by the following system of differential equations for two species x and y: $\frac{dx}{dt} = x(a_x - b_x y) \\ \frac{dy}{dt} = y(c_y - d_y x)$ Here, \(a_x\) is the intrinsic growth rate of species x, \(b_x\) is the competition coefficient affecting species x by species y, \(c_y\) is the intrinsic death rate of species y, and \(d_y\) is the competition coefficient affecting species y by species x.

Find the equilibrium points

To find the equilibrium points, we need to solve for x and y when the populations are not changing, i.e., when \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\) Setting each of the equations equal to zero: $x(a_x - b_x y) = 0 \\ y(c_y - d_y x) = 0$ Next, we solve for x and y.

Solve for the equilibrium points

From the above equations, we can derive four equilibrium points: 1. \(x = 0, y = 0\): Both species are extinct. 2. \(x = 0, y = \frac{c_y}{d_y}\): Species x is extinct, and species y is at its carrying capacity. 3. \(x = \frac{a_x}{b_x}, y = 0\): Species y is extinct, and species x is at its carrying capacity. 4. \(x = \frac{a_x}{b_x}, y = \frac{c_y}{d_y}\): Both species coexist. We are interested in the fourth case, where both species coexist.

Identify the necessary condition for coexistence

To have stable coexistence, the following condition should be met: 1. The competition coefficients should be less than the intrinsic growth/death rates, that is: \(b_x < a_x\) and \(d_y < c_y\) This condition is necessary for both species to coexist because it ensures that the population growth of one species is not suppressed too much by the presence of the other species. To sum up, the necessary condition for coexistence in the Lotka-Volterra competition model is that the competition coefficients must be less than the intrinsic growth/death rates of the respective species: \(b_x < a_x\) and \(d_y < c_y\)

Key concepts

Interspecific Competition

Interspecific competition occurs when different species vie for the same resource that is in limited supply. This struggle can influence various aspects of the species' lives, including their distribution, abundance, and evolutionary adaptations. In the context of the Lotka-Volterra model, this competition is quantified through competition coefficients in the differential equations that describe the population dynamics.

The coefficients, such as \(b_x\) and \(d_y\), represent the intensity of the impact that one species has on another. For example, if species y has a high competition coefficient \(b_x\), it means that as the population of species y increases, it will greatly reduce the growth rate of species x. Conversely, a lower coefficient indicates less impact. These interactions are central to predicting long-term population trends and understanding how species can coexist or outcompete each other in an ecosystem.

Equilibrium Points

Equilibrium points in a biological sense refer to the population sizes of each species where no further changes occur — that is, where the populations are stable. In the Lotka-Volterra competition model, these points are found by setting the rate of change in the population to zero. Mathematically, this equates to solving the system of equations \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\).

Through careful analysis, as demonstrated in the exercise, we identify conditions under which populations will stabilize. Either one species dominates, the other goes extinct, or they coexist at a balance. These insights provide crucial information about the sustainability of ecosystems and how different species can impact one another's survival. In real-world applications, this helps in conservation efforts and in managing natural resources.

Species Coexistence

Coexistence of multiple species in the same environment is a delicate balance that is deeply influenced by interspecific competition. According to the Lotka-Volterra competition model, stable coexistence is possible under specific conditions. Specifically, each species' competition effect on the other must be weaker than the intrinsic growth or death rates, indicated by the inequalities \(b_x < a_x\) and \(d_y < c_y\).

This condition essentially means that although there is competition between the species, it is not so intense that it leads to the extinction of one species. Instead, both species exert limited pressure on one another, allowing them to persist together in the ecological community. Understanding these conditions is critical for predicting the outcomes of species introductions or the impacts of environmental changes on species diversity.