Impala Management

(From Mooney and Swift, A Course In Mathematical Modeling.)

Background and Problem

The purpose of this project is to produce a report examining the effects of several different types of game management on the population dynamics of an impala herd. In particular, it should report on herd population in impala are managed by (1) predation, (2) trophy hunting, or (3) game ranch management. For each of these three cases it will consider the effects of low levels of hunting or predation (6% of the population) and high levels of hunting or predation (16% of the population). In all, the report should compare the effects of six management strategies on the population over time. Aside from the data and definitions below, you are on your own to construct the appropriate model and run it for an appropriate length of time.

Definitions and Data

All data is taken from [GM].

Predation means male and female impala are killed equally and 45% of the kill is juvenile, 20% of the kill is sub-adult and 35% of the kill is adult.

Trophy hunting means 70% of the impala killed are male and of all the animals killed 2.5% are juvenile, 2.5% are sub-adult, and 95% are adult. Basically the old males with horns are the desired game.

Game ranching means 70% of the impala killed are males, but the age proportions are 5% juvenile, 75% sub-adult, and 20% adult. Here the desired product is meat. More males are killed because the females give birth.

For example, if we were game ranching at a high level and the population was at 100, we would kill 16 animals. Of those 16, 11 (actually, 11.2) would be male, and 5 (actually 4.8) would be female. Of the males, 1 (actually 0.56) would be juvenile, 8 (actually 8.4) would be sub-adult, and 2 (actually 2.24) would be adults. Of the females, 0 (actually 0.25) would be juvenile, 4 (actually 3.6) would be sub-adult, and 1 (actually 0.96) would be adult.

Impala live 11 years. A juvenile is less than one year old, a sub-adult is at least one year old and less than 5 years old, and an adult is 5 years old or older. Below is a table of survival and reproduction rates for impala, in the absence of hunting. The survival rates give the percentage of members of that age class who survive to enter the next age class.

Age Class	Male survival rates	Female survival rates	Birth rates (females born to females)
1	0.60	0.60	0
2	0.80	0.90	0.35
3	0.95	0.95	0.45
4	1.00	0.97	0.45
5	1.00	0.97	0.45
6	1.00	0.95	0.45
7	1.00	0.95	0.45
8	0.75	0.95	0.45
9	0.34	0.70	0.45
10	0	0.80	0.45
11	0	0	0.45

To simplify matters, work with partial animals (e.g. 3.25 impala), and take sub-adults (and adults) equally from the various age classes. If hunting at a certain level wipes out an age class, just kill all the animals within it and don't worry about the fact that fewer than 6% or 16% are removed that year. Be careful not to create negative animals!

Fifty percent of the impala born are female, and fifty percent are male.

Only adult males breed, and one male can breed with three females. If the population of breeding males drops below a third of the breeding females, then the female birth rates must be adjusted downward to compensate. Assume that downward compensation follows a linear relation with birth rates of 0 if no males are present.

Assume to start with that there are 220 animals equally distributed among age classes and sexes. Let your model run for 15 years without management to stabilize the natural growth rate. This will ensure that the system is close to some equilibrium state before it is perturbed by management, and this will eliminate the effects of the choice of the initial population distribution. it will also tell how an unmanaged herd will grow, which will give a baseline to compare with the management results.

It might be advisable to start with as simple a model as possible and then add complications. To deal with the two sexes, use a single population vector with 22 entries, where the first 11 entries are for the females and the next 11 are for the males (alternatively, you could interleave them: females in group 1, males in group 1, females in group 2, males in group 2, etc.).

References

[GM] J.R. Ginsberg and E.J. Milner-Gulland. "Sex biased harvesting and population dynamics in ungulates: Implications for conservation and sustainable use." Conservation Biology, vol. 8, no. 1, pp. 157-166, 1994