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Chapter 7: Problem 11

Consider the system of equations $$ \begin{array}{l} \frac{d x}{d t}=k_{1} x\left(1-\frac{x}{L_{1}}\right)-a x y \\ \frac{d y}{d t}=k_{2} y\left(1-\frac{y}{L_{2}}\right)-b x y \end{array} $$ where \(x(t)\) and \(y(t)\) give the populations of two species \(A\) and \(B\), respectively, and \(k_{1}, k_{2}, L_{1}, L_{2}, a\), and \(b\) are positive constants. a. Describe what happens to the population of \(A\) in the absence of \(B\). b. Describe what happens to the population of \(B\) in the absence of \(A\). c. Give a physical interpretation of the roles played by the terms \(a x y\) and \(b x y\), and explain why the equations are called competing species equations. (Examples of competing species are trout and bass.) d. Find the equilibrium points and interpret your results.

Short Answer

Expert verified
In the absence of B, the population of A follows logistic growth until it reaches carrying capacity \(L_1\), and similarly, in the absence of A, the population of B follows logistic growth until it reaches carrying capacity \(L_2\). The terms \(axy\) and \(bxy\) represent the competition between species A and B for limited resources, leading to the name "competing species equations." There are three equilibrium points: (1) both species are extinct, (2) species A has reached its carrying capacity and species B is extinct, and (3) species B has reached its carrying capacity and species A is extinct. Additional equilibrium points may exist depending on the values of constants \(k_1, k_2, L_1, L_2, a\), and \(b\).

Step by step solution

01

a. Population of A in the absence of B

In the absence of B, the population of B would be zero (\(y = 0\)). So, the second term in the equation for \(\frac{dx}{dt}\) would be zero, and the equation becomes: $$ \frac{dx}{dt} = k_1 x \left(1-\frac{x}{L_1}\right). $$ The above equation describes the growth of species A in the absence of species B. The growth would follow the logistic growth model until it reaches the carrying capacity \(L_1\).
02

b. Population of B in the absence of A

Similarly, in the absence of A, the population of A would be zero (\(x = 0\)). So, the second term in the equation for \(\frac{dy}{dt}\) would be zero and the equation becomes: $$ \frac{dy}{dt} = k_2 y \left(1-\frac{y}{L_2}\right). $$ The above equation describes the growth of species B in the absence of species A. The growth would also follow the logistic growth model until it reaches the carrying capacity \(L_2\).
03

c. Physical interpretation and competing species equations

The terms \(axy\) and \(bxy\) signify the interaction between the two species A and B. They represent the competition between the species for some limited resource (such as food or space). The more they interact, the more their population growth is affected. The negative sign in front of the terms (\(-axy\) and \(-bxy\)) implies that the presence of one species has a negative effect on the growth rate of the other species. This is why the equations are called "competing species equations." Examples of competing species could be trout and bass, which compete for similar resources in water bodies.
04

d. Equilibrium points and interpretation

To find the equilibrium points, we set both \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) to zero and solve for \(x\) and \(y\). So, we get: $$ \begin{array}{l} k_1 x\left(1-\frac{x}{L_1}\right) - a x y = 0, \\ k_2 y\left(1-\frac{y}{L_2}\right) - b x y = 0. \\ \end{array} $$ There are three equilibrium points: \((0, 0)\), \(\left(L_1, 0\right)\), and \(\left(0, L_2\right)\). 1. \((0, 0)\): This point represents the case when both species are extinct. There are no interactions, so no effect on each other's growth. 2. \((L_1, 0)\): At this point, species A has reached its carrying capacity, while species B is absent (extinct). In this state, species A can grow freely without any competition from species B. 3. \((0, L_2)\): This is the reverse of the previous case. Species B has reached its carrying capacity, while species A is extinct. In this state, species B can grow freely without any competition from species A. Other nontrivial equilibrium points may also exist, depending on the specific values of the constants \(k_1, k_2, L_1, L_2, a, b\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logistic Growth Model
The Logistic Growth Model is used to describe how a population grows in an environment with limited resources.
The formula for logistic growth is given by: \[\frac{dN}{dt} = rN\left(1-\frac{N}{K}\right),\]where:
  • \( N \) is the population size at time \( t \).
  • \( r \) is the intrinsic growth rate of the population.
  • \( K \) is the carrying capacity or maximum population size that the environment can sustain.
As a population approaches its carrying capacity, the growth rate slows due to limited resources.
This results in an "S-shaped" curve when population size is plotted against time.
In the given competing species model, each species (A and B) tends to follow a logistic growth pattern when isolated.
However, they are limited by their respective carrying capacities \( L_1 \) and \( L_2 \) respectively.
Equilibrium Points
Equilibrium points are where the growth rates of species reach zero, meaning the population sizes do not change over time.
To find these points, we set the growth equations to zero: \[k_1 x\left(1-\frac{x}{L_1}\right) - ax y = 0, \]and \[k_2 y\left(1-\frac{y}{L_2}\right) - bxy = 0.\]The solutions give us conditions under which populations remain unchanged.
For the system of two species, there are several essential equilibrium points:
  • \( (0, 0) \) - Both species A and B are extinct.
  • \( (L_1, 0) \) - Species A reaches its carrying capacity, while species B is extinct.
  • \( (0, L_2) \) - Species B reaches its carrying capacity, while species A is extinct.
The existence of these points indicates stable or unstable conditions that would impact how populations interact over time.
Species Interaction
Species interaction in a competing species model is characterized by the terms that involve both species' populations.
In the equations provided, the terms \(-ax y\) and \(-bxy\) represent these interactions.
  • \( axy \) indicates species A's negative impact on species B's growth rate.
  • \( bxy \) indicates species B's negative impact on species A's growth rate.
The negative sign before the interaction terms suggests competition.
Each species hinders the other's growth when they share an ecosystem.
This competition typically revolves around resources such as food, light, or space.
When resources are scarce, populations tend to decrease because each species interferes with the other's access to these resources.
Population Dynamics
Population dynamics examines how populations change over time and space, influenced by factors like birth rates, death rates, and species interactions.
In a competing species model, dynamics can be complex, reflecting both the logistic growth and interactions.
  • Logistic growth allows populations to flourish until they approach carrying capacity, slowing due to limited resources.
  • Interaction terms \(-axy\) and \(-bxy\) alter these dynamics by introducing competition.
Overall, the population dynamics in competing species models are influenced by both individual species' growth tendencies and their interactions.
Unique outcomes could emerge based on parameters like growth rates \( k_1 , k_2 \), interaction coefficients \( a, b \), and carrying capacities \( L_1, L_2 \).
Understanding these dynamics helps in predicting the behavior of species in natural ecosystems.

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