interspecific competition between two species

Interspecies competition
　Organisms compete for various resources. If these resources are also used by other species and the populations of other species fluctuate, how should we express them in a model of population dynamics (community dynamics) of various species?
　The following Lotka-Volterra interspecific competition model is a model for expressing the population dynamics of two species (referred to here as species 1 and species 2) in a situation where they compete for resources.

\( \frac{dN_1}{dt} = r_1N_1 \frac{K_1 - N_1 - \alpha_{12}N_2 }{K_1} \)    (6.5)

\( \frac{dN_2}{dt} = r_2N_2 \frac{K_2 - N_2 - \alpha_{21}N_1 }{K_2} \)    (6.6)

This model is a system of differential equations. Equation 6.5 represents the population dynamics of species 1, and Equation 6.6 represents the population dynamics of species 2. Species 1 and 2 are distinguished by subscripts 1 and 2 in equations 6.5 and 6.6 because they differ in population size, internal natural increase rate, and carrying capacity. For example, N₁ and r₁ are the population and internal natural increase rate of species 1. This model looks complicated at first glance, but it's actually a formula based on a simple idea, so let's think about it carefully without showing rejection.
　First, Equation 6.5, which expresses the population dynamics of species 1, will be explained. Equation 6.5 is basically the same as Equation 6.4, but a new term –α₁₂ N₂ is added to the last numerator. α₁₂ is called the competition coefficient. Without species 2, the growth rate of species 1 would be dN₁/dt = 0 when K₁ – N₁ = 0. However, when species 2 is present, interspecific competition occurs, and species 2 acquires resources that species 1 was supposed to use. This interspecific competition affects the growth rate of species 1, similar to the density effect due to intraspecific competition. However, since Species 1 and 2 are different organisms, their use of resources and superiority in competition are different. Therefore, the competition coefficient α₁₂ expresses how many individuals of species 1 correspond to one species of species 2. For example, if α₁₂ = 2, then one individual of species 2 reduces the carrying capacity of species 1 by two individuals of species 1. If α₁₂ = 0.5, two individuals of species 2 are one individual of species 1. The greater the population of Species 2, the greater the carrying capacity of Species 1 is reduced by Species 2. In this way, Equation 6.5 is a model in which the population dynamics of species 1 are affected not only by the number of individuals of species 1, but also by the product of the number of individuals of species 2 and the competition coefficient.
　On the other hand, Equation 6.6 represents the population dynamics of species 2. The competition coefficient α₂₁ represents how many individuals of species 2 correspond to one individual of species 1. In this way, using the concept of the competition coefficient, even in situations where there is interspecific competition among m species, the negative impact on species 1 can be expressed as α₁₂ N₂, α₁₃ N₃, ..., α_1m N_m. However, such calculations are very difficult.
　Now, what kind of behavior will the populations of the two species exhibit by combining Equations 6.5 and 6.6? The relationship between the populations of the two species and time changes in various ways depending on the value of each variable. Because species 1 has a higher internal natural increase rate r₁, it increases in population faster than species 2, but later populations of one of the two species may become extinct or coexist.
The population dynamics of two species leading to extinction or coexistence does not necessarily mean that the weaker species will unilaterally decline or that the species that increases first will continue to dominate. Species that flourished earlier may eventually become extinct. To get a feel for this diverse population dynamics, it is easy to simulate with a computer.

isocline method
　Even without simulations, if you just want to conclude what kind of equilibrium state the two species will eventually be in, you can study it using a diagram using a technique called the isocline method. The equilibrium state of formulas 6.5 and 6.6 refers to a state in which the growth rates of the two species (dN₁/dt, dN₂/dt) become zero. By inserting 0 into the left side of Equations 6.5 and 6.6 and transforming the equations, the following simple equations are obtained.

N₁ = K₁ – α₁₂N₂　　　（6.7）

N₂ = K₂ – α₂₁N₁　　　（6.8）

Since carrying capacity and competition coefficient are constants, they are both linear functions of N₁ and N₂. Let's plot Eqs. 6.7 and 6.8 for X > 0 and Y > 0, with the population of species 1 on the x-axis and the population of species 2 on the y-axis. Four different figures are drawn with two straight lines.

(a) The species 1 line is generally above the species 2 line.
(b) The species 2 line is generally above the species 1 line.
(c) When the two lines intersect and the population of species 1 is small, the line for species 2 is higher than the line for species 1, and when the population of species 1 is greater than the population of the intersection, the line for species 1 is at the top.
(d) When the two lines intersect and the population of species 1 is small, the line for species 1 is higher than the line for species 2, and when the population of species 1 is greater than the population of the intersection, the line for species 2 is at the top.

On the straight line drawn by these two functions, the zero growth line (zero isocline), the population of each species is in a steady state, neither increasing nor decreasing. However, as the distance from the straight line increases, the absolute value of the growth rate (dN₁/dt or dN₂/dt) of species 1 and 2 increases, so the population increases or decreases at a faster rate.
　Let's take a closer look at the diagram. The populations of the two species continue to fluctuate over time, but eventually reach equilibrium. The above four diagrams are in the following equilibrium states in order.

(a) If the line for species 1 is entirely above the line for species 2, then only species 1 will survive and species 2 will become extinct (zero population). Even if species 2 reappeared, species 2 would go extinct again. This is a stable equilibrium state. When the population of species 1 is 0, it is also in equilibrium, but if there is even a little seed 1, the number of seed 1 will increase, so this is an unstable equilibrium state.

(b) Thinking the same way, only species 2 survive.

(c) The intersection of the two straight lines is the unstable equilibrium. In other words, if the populations of the two species remain at the point of intersection, equilibrium will be maintained, but if the coordinates of the intersection point are deviated even slightly, the populations of the two species will move away from the point of intersection over time, eventually reaching a stable equilibrium in which only one species survives.

(d) The point of intersection of the two straight lines is the stable equilibrium state. In other words, of (a) to (d), only (d) is where the two species coexist stably.

　From the results of such studies, we can see the necessary and sufficient conditions for the coexistence of the two species to be in a stable equilibrium state. The condition is that if we focus on the intercepts of the Y-axis and the X-axis,

K₁ < K₂/α₂₁  ∧　K₂ < K₁/α₁₂

The biological meaning of this formula can be easily understood by transforming it as follows.

1/K₁ < α₂₁/K₂  ∧　1/K₂ < α₁₂/K₁

In other words, the conditions necessary for the coexistence of two species are (1) the density effect (1 / K₁) of one individual of species 1 on the population of the same species is greater than the effect of that individual on species 2 (α₂₁ / K₂) and (2) the influence of one individual of species 2 on the conspecific population (1 / K₂) is greater than the influence of that individual on species 1 (α₁₂ / K₁).

In short, two species coexist when their influence on the same species is stronger than on the other.
　Consider the hermit crab, for example. Many species of hermit crabs coexist on Japanese coasts. Each species of hermit crab uses one shell. Therefore, the competition coefficients α₁₂ and α₂₁ appear to be 1. However, if their distributions and preferences for shell resources are partially different, and as a result, there is a difference in the use of shells, the competition coefficient will be less than 1. On the rocky shores of Akkeshi Bay in eastern Hokkaido, Pagurus middendorffii and Pagurus brachiomastus coexist.

These two species of hermit crabs have distinct distributions and preferences for shell resources (Oba et al. 2008). Such resource division may allow hermit crabs to meet the conditions for coexistence.