In the classic Lotka-Voltetta equations, the prey grow exponentially and are kil
ID: 3005343 • Letter: I
Question
In the classic Lotka-Voltetta equations, the prey grow exponentially and are killed by predators according to a type I functional response. Rewrite these differential equations assuming that predators kill prey according to a type II functional response but maintain exponential growth.
dN dt = rN bNP
dP dt = cbNP dP
type 1= bN type 2= aN/b+N
Solve for the equilibrium population of both predators and prey. This requires setting both predator and prey equations to zero, which you did to solve for the nullclines, and then solving the two nullcline equations using the method of substitution. Hint: One of your nullclines already tells you the equilibrium population.
Explanation / Answer
Population equlibrium occurrs when popuation of both are not chnaging that is derivaties are zero. Note that P denotes predoator and N is the prey.
dN/dt = 0 => rN - aN/(b+N)P = 0 (1)
dP/dt = 0 => caN/(b+N)P -dP = 0 (2)
From (1) , N( r - a/(b+N)P) = 0, or, r = aP/b+N => P = r(b+N)/a (3)
Once we find N from (2), we can find the equlibrium population of P
From (2), P( caN/b +N - d) = 0, or d = caN/b + N
=> bd + dN = caN . Re-arranging, N (ca - d) = bd or N = bd/ (ca - d)
Putting above in (3), P = r ( b + bd/ ca - d)/ a = (r/a) (abc - bd + bd)/ca - d = rbc/ca -d. Therefore P = r bc/(ca - d)