17 Two similar species, X and Y , of a small mammal compete for food and habitat. A model of this competition assumes, in a particular area, the following.
- In the absence of the other species, each species would increase at a rate proportional to the number present with the same constant of proportionality in each case.
- The competition reduces the rate of increase of each species by an amount proportional to the number of the other species present.
So if the numbers of species X and Y present at time \(t\) years are \(x\) and \(y\) respectively, the model gives the differential equations
\(\frac { d x } { d t } = k x - a y\) and \(\frac { d y } { d t } = k y - b x\),
where \(k , a\) and \(b\) are positive constants.
- Show that the general solution for \(x\) is \(x = A e ^ { ( k + n ) t } + B e ^ { ( k - n ) t }\), where \(n = \sqrt { a b }\) and \(A\) and \(B\) are arbitrary constants.
- Hence find the general solution for \(y\) in terms of \(A , B , k , n , a\) and \(t\).
Observations suggest that suitable values for the model are \(k = 0.015 , a = 0.04\) and \(b = 0.01\). You should use these values in the rest of this question.
- When \(t = 0\), the numbers present of species X and Y in this area are \(x _ { 0 }\) and \(y _ { 0 }\) respectively.
- Show that \(\mathrm { x } = \frac { 1 } { 2 } \left( \mathrm { x } _ { 0 } - 2 \mathrm { y } _ { 0 } \right) \mathrm { e } ^ { 0.035 \mathrm { t } } + \frac { 1 } { 2 } \left( \mathrm { x } _ { 0 } + 2 \mathrm { y } _ { 0 } \right) \mathrm { e } ^ { - 0.005 \mathrm { t } }\).
- Hence show that \(y = \frac { 1 } { 4 } \left( x _ { 0 } + 2 y _ { 0 } \right) e ^ { - 0.005 t } - \frac { 1 } { 4 } \left( x _ { 0 } - 2 y _ { 0 } \right) e ^ { 0.035 t }\).
- Use initial values \(x _ { 0 } = 500\) and \(y _ { 0 } = 300\) with the results in part (b) to determine what the model predicts for each of the following questions.
- What numbers of each species will be present after 25 years?
- In this question you must show detailed reasoning.
When will the numbers of the two species be equal?
- Does either species ever disappear from the area? Justify your answer.
- Different initial values will apply in other areas where the two species compete, but previous studies indicate that one species or the other will eventually dominate in any given area.
- Identify a relationship between \(x _ { 0 }\) and \(y _ { 0 }\) where the model does not predict this outcome.
- Explain what the model predicts in the long term for this exceptional case.