This non-dimensionalized, sleekest most neatest model illustrates predator prey interactions using logistic growth for the moose population, for the wolf and moose populations on Isle Royale.
Thanks Scott Fortmann-Roe for the original model.
I've added in an adjustment to handle population sizes, by dividing by moose carrying capacity.
Time is scaled by the moose birth parameter:
There are therefore only three parameters left to account for any dynamics:
beta = bw/bm (relative wolf to moose births)
delta = dm/bm (relative death to birth ratio for moose)
gamma = dw/bm (wolf deaths to moose births)
The equations are thus
dM/dtau = M [ (1-M) - delta W ]
dW/dtau = W [beta M - gamma ]
There is a stable equilibrium pair of population values, relative to the carrying capacity:
M^* = gamma / beta
W^* = (1-gamma / beta) / delta
I have a sleek version with a logistical growth term for the moose, athttp://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/Moose-n-Wolf-InsightMaker-sleek.nb