Find all of the equilibrium points for two systems. Explain the significance of these points in terms of the predator and prey populations dxdt=10x(1−x10)−20xy,dydt=−5y+xy20 (i) dxdt=0.3x−xy100,dydt=15y(1−y15)+25xy (ii)

Question

Find all of the equilibrium points for two systems. Explain the significance of these points in terms of the predator and prey populations  dxdt=10x(1−x10)−20xy,dydt=−5y+xy20   (i)  dxdt=0.3x−xy100,dydt=15y(1−y15)+25xy   (ii)

Aamina Herring · Accepted Answer

For equilibrium point, dxdt=0 and dydt=0  For system (i): dxdt=0  10x(1−x10)−20xy=0  x(10−x−20y)=0 .....(1) dydt=0  −5y+xy20=0  y(−5+x20)=0......(2)  Solution of the equations (1) and (2) will have 3 cases. Case 1:  x=0,y=0  This will represent 0 prey and 0 predator.  Case 2:  −5+x20=0  x=100  10−x−20y=0  10−100−20y=0  y=−4.5  As y cannot be negative, the solution set is rejected.  Case 3:  when y=0  10−x−20(0)=0  10−x=0  x=10  For system (ii):  dxdt=0  0.3x−xy100=0  x(0.3−y100)=0 .....(3)  dydt=0  15y(1−y15)+25xy=0  y(15−y+25x)=0 ...(4)  Solution of the equations (3) and (4) will have 3 cases.  Case 1:  x=0,y=0  This will represent 0 prey and 0 predator.  Case 2:  0.3−y100=0  y=30  15−y+25x=0  15−30+25x=0  x=35  So, (35,30) will represent a stable population.  Case 3:  when y=0  15−(0)+25x=0  x=−35  As x cannot be negative, the solution set is rejected.

Find all of the equilibrium points for two systems. Explain the significance of these points in terms of the predator and prey population

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