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Chapter 3: Problem 7

Competition between populations: In this exercise we consider the problem of competition between two populations that vie for resources but do not prey on each other. Let \(m\) be the size of the first population, let \(n\) be the size of the second (both measured in thousands of animals), and assume that the populations coexist eventually. An example of one common model for the interaction is Per capita growth rate for \(m\) is \(3(1-m-n)\) Per capita growth rate for \(n\) is $$ 2(1-0.7 m-1.1 n) $$ At an equilibrium point the per capita growth rates for \(m\) and for \(n\) are both zero. If the populations reach such a point, then they will continue at that size indefinitely. Find the equilibrium point in the example above.

Short Answer

Expert verified
The equilibrium point is \(m = 0.25\) (250 animals) and \(n = 0.75\) (750 animals).

Step by step solution

01

Set the Growth Rates to Zero

At equilibrium, the per capita growth rates for both populations are zero. For population \(m\), the per capita growth rate equation is \(3(1-m-n)\). Setting it to zero gives: \[3(1 - m - n) = 0\] Solving this equation will give us one condition for equilibrium.
02

Solve for Population m

Simplify the equation from Step 1: \[3 - 3m - 3n = 0\] Divide through by 3 and rearrange to solve for \(m\): \[m + n = 1\]
03

Set the Second Growth Rate to Zero

For population \(n\), the per capita growth rate equation is \(2(1-0.7m-1.1n)\). Setting it to zero gives: \[2(1 - 0.7m - 1.1n) = 0\] Simplify to establish the second equilibrium condition.
04

Solve for Population n

Simplify the equation from Step 3: \[2 - 1.4m - 2.2n = 0\] Divide through by 2, rearrange and solve for \(n\): \[1 - 0.7m - 1.1n = 0\] Rewriting gives: \[0.7m + 1.1n = 1\]
05

Simultaneous Equations

From Steps 2 and 4, we have two equations: 1. \(m + n = 1\) 2. \(0.7m + 1.1n = 1\) Solve these equations simultaneously to find the values of \(m\) and \(n\).
06

Solve for n Using Substitution

From \(m + n = 1\), we have \(m = 1 - n\). Substitute this into the second equation: \[0.7(1-n) + 1.1n = 1\] Simplify and solve for \(n\): \[0.7 - 0.7n + 1.1n = 1\] \[0.4n = 0.3\] \[n = \frac{0.3}{0.4} = 0.75\]
07

Solve for m

Substitute \(n = 0.75\) back into \(m = 1 - n\): \[m = 1 - 0.75 = 0.25\]
08

Conclusion

The equilibrium point, where both populations' per capita growth rates are zero, is \((m, n) = (0.25, 0.75)\). This means 250 animals in population \(m\) and 750 animals in population \(n\) at equilibrium.

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Key Concepts

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

Understanding Competitive Populations
Competitive populations are groups of species that are vying for the same resources, such as food, water, or territory, but do not engage in predation on each other. In our case, two species share a habitat and have to balance their growth based on the availability of these shared resources. When both populations reach a point where their growth does not change (i.e., an equilibrium point), they are said to coexist in a sustainable manner.

At equilibrium, each population has adjusted its growth rate so that it neither increases nor decreases in size. This stable state is essential for understanding how species coexist without one outcompeting the other completely. Models like these help ecologists predict the outcomes of resource sharing and enable better resource management strategies.
Per Capita Growth Rate Explained
The per capita growth rate is a measure of how much each individual in a population contributes to the overall growth of that population. It essentially reflects the net rate of increase per individual unit or animal in a specific time period.

Mathematically, in our example, the initial per capita growth rate for population \(m\) is given by \(3(1-m-n)\) and for population \(n\) by \(2(1-0.7m-1.1n)\). This tells us how the size of each population influences its own growth.

When resource competition is factored in, the per capita growth rate becomes zero at equilibrium, indicating that the population size remains constant. This zero growth rate means that all births and incoming resources balance out with deaths and resource consumption, establishing a steady state where populations can sustainably coexist.
Solving with Simultaneous Equations
Simultaneous equations are a set of equations with multiple variables which must be solved at the same time. In this scenario, they help determine the equilibrium point where both populations stop growing.

For our populations, we derived two equilibrium conditions: \(m + n = 1\) and \(0.7m + 1.1n = 1\). Solving these equations together allows us to find the exact values of \(m\) and \(n\) that satisfy both conditions simultaneously. Solving involves either the substitution or elimination method to find exact numerical outcomes for the variables.

In our case, substituting \(m = 1 - n\) into the second equation, we simplified and solved to find \(n = 0.75\). Subsequently, substituting \(n\) back gives \(m = 0.25\). These values represent the equilibrium point, ensuring that both populations cease growth while efficiently sharing their environment's resources.

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