Question

In the study of ecosystems, predator-prey models are often used to study the interaction between species. Consider populations of tundra wolves, given by \(W\left( t \right),\) and caribou, given by \(C\left( t \right),\) in northern Canada. The interaction has been modeled by the equations

\(\frac{{dC}}{{dt}} = aC - bCW\,\,\,\,\,\,\,\,\,\frac{{dW}}{{dt}} = - cW + dCW\)

1. What value of \(\frac{{dC}}{{dt}}\) and \(\frac{{dW}}{{dt}}\) corresponds to a stable populations?
2. How would the statement “The caribou go extinct” be represented mathematically?
3. Suppose that \(a = 0.05,b = 0.001,c = 0.05,\) and \(d = 0.0001\). Find all population pairs \(\left( {C,W} \right)\) that lead to stable populations. According to this model, is it possible for the two species to live in balance or will one or both species become extinct?

Short answer

a)The values are \(\frac{{dC}}{{dt}} = 0\) and \(\frac{{dW}}{{dt}} = 0\) correspond to stable populations.

b)The mathematical representation of the statement is \(C = 0\).

c) The population pairs \(\left( {C,W} \right)\), which yield stable populations, are \(\left( {0,0} \right)\) and \(\left( {500,50} \right)\). It is possible for both species to live together peacefully.

Step-by-step solution

Determine the values of \(\frac{{dC}}{{dt}}\) and \(\frac{{dW}}{{dt}}\) corresponds to stable populations

a)

When there is a stable population then the growth rate is neither positive nor negative. In other words \(\frac{{dC}}{{dt}} = 0\) and \(\frac{{dW}}{{dt}} = 0\).

Thus, the values are \(\frac{{dC}}{{dt}} = 0\) and \(\frac{{dW}}{{dt}} = 0\) correspond to stable populations.

The mathematical representation of the statement

b)

The statement “The caribou go extinct” signifies that the population has reached zero. The mathematical representation of the statement is \(C = 0\).

Determine all population pairs \(\left( {C,W} \right)\) that lead to a stable population

c)

It is given that the two equations are \(\frac{{dC}}{{dt}} = aC - bCW\) and \(\frac{{dW}}{{dt}} = - cW + dCW\).

Consider that \(\frac{{dC}}{{dt}} = 0\), \(\frac{{dW}}{{dt}} = 0\), \(a = 0.05,b = 0.001,c = 0.05,\) and \(d = 0.0001\) to obtain the equations as shown below:

\(0.05C - 0.001CW = 0\)……(1)

\( - 0.05W + 0.0001CW = 0\)……(2)

Add 10 times equation (2) to (1) removes the \(CW\) terms to obtain as shown below:

\(0.05C - 0.5W = 0\)

Substitute \(C = 10W\) in equation (1) as shown below:

\(\begin{array}{c}0.05\left( {10W} \right) - 0.001\left( {10W} \right)W = 0\\0.5W - 0.01{W^2} = 0\end{array}\)

Multiply the above equation by 100 as shown below:

\(\begin{aligned}50W - {W^2} & = 0\\W\left( {50 - W} \right) & = 0\\W = 0\,\,\,{\mathop{\rm or}\nolimits} \,\,\,W & = 50\end{aligned}\)

Substitute \(W = 0\,\,\,{\mathop{\rm and}\nolimits} \,\,W = 50\) in \(C = 10W\) to obtain as shown below:

\(\begin{aligned}C & = 0\\C & = 10\left( {50} \right)\\ & = 500\end{aligned}\)

Therefore, the population pairs \(\left( {C,W} \right)\), which yield stable populations, are \(\left( {0,0} \right)\) and \(\left( {500,50} \right)\). Therefore, both species can live together.

Thus, it is possible for two species to live together.