Many other models fall into a category that can broadly be described as ecological models, The most famous example is the simple two-species predator-prey model of Lotka and Volterra (Lotka (1925) and Volterra (1926), see also Roughgarden (1979), Murray (1990)):
dV dt = V(b V ;d V P) (1.11) dP dt = P(b P V ;d P)
HereV is the prey (victim) population size,P the predator population size, and the total population
size V +P is not xed. Without predator-prey interactions (which are again assumed to occur at
a rate proportional to V P using the mass-action principle) the prey, feeding on an unlimited
resource, grow exponentially at their birthrateb
V, and the predators die from starvation at rate d
P.
Prey numbers are directly checked by predation with deaths occurring at rated V
VP, and new prey
are born at a similar rateb V
VP more indirectly due to the increased food supply for the predators.
Equations (1.11) famously exhibit cycles (gure (1.4)) in which the number of predators endlessly chase the number of prey. The equations are neutrally stable and no matter what initial conditions, providedP and V are non-zero, the populations will both keep returning to their starting values
simultaneously. (The functionH, dened by H=b P V +d V P;d Plog b P d P V ;b V log d V b V P
is found to be constant in time.) There is one internal equilibrium atV =b V =d V and P =d P =b P
which is also neutrally stable.
More generally, Lotka-Volterra type interactions can be extended to an arbitrary number of species,
n, by considering the equations dx i dt =x i 0 @ r i+ n X j=0 A ij x j 1 A (1.12) where x
i is the number of species
i andr i and
A
ij are constants. The intrinsic rates of increase r
i
can be either positive for birth rates (as for the prey above) or negative for death rates (as for the predators) the interaction constants A
ij represent in a non-mechanistic way the eect of species j
0 500 1000 1500 2000 2500 3000 3500 4000 0 100 200 300 400 500 600 700 800 900 1000 prey V predator P
Figure 1.4: Examples of the cycles in phase space produced by the Lotka-Volterra predator prey model. Four dierent initial conditions are shown, leading to four dierent closed orbits which ow anti-clockwise. The internal equilibrium is marked with a cross. The parameters areb
V = 1 :0b P= 0 :005d V = 0 :01 andd P = 5 :0.
represent harmful eects (e.g. parasitism, predation, competition for resources) and positive ones
are benecial (catching food, symbiosis), so these equations can be formulated to describe a wide range of species interactions and interdependencies, albeit non-mechanistically.
One should note the similarity between the game theory equations (1.10) and equations (1.12). Both determine species abundances by a payo or inuence matrix the rst in terms of frequency, the second by actual population size.
Such models are clearly just about as simple as it is possible to get when considering interacting species, but it is possible to add further realism. Density-dependent growth is one way, e.g. by
replacing a birth rater xwith logistic growthr x(N;x). If this is done for the predator-prey model,
all the non-generic limit cycles collapse to one globally attracting (in the positive quadrant) xed point of coexistence, providingN is suciently large (greater than d
P =b
P). If
N is too small, the
predators cannot coexist with the prey and are eliminated altogether. A further natural modication is predator satiation, where the mass-action assumptionVP is modied to take account of the fact
that a xed number of predators can't kill prey at arbitrarily high rates. The linear dependence on
V inVP is replaced by a term that is also increasing, but is bounded above, for examplec(1;e ;k V)
with c andk constants. Including this with prey density-dependence gives a system of equations
which either collapse to an internal equilibrium or undergo a Hopf bifurcation and exhibit a unique attracting limit cycle towards which all trajectories tend. Time delays are another possible addi-
tion, perhaps most naturally expressed in the predator-prey system through predator birth being dependent on prey availability some time in the past,b
P
V(t;T)P(t), instead of at current levels, b
P
V(t)P(t) thus complicating any analysis considerably. Nevertheless, useful analytical results are
available in some cases. Lyapunov functions, for example, are a valuable technique for proving the global asymptotic stability of equilibria, as in Ardito and Ricciardi (1995) who study a wide class of predator-prey systems.
In this manner, more and more improvements to the basic model can be made, generally in a non- mechanistic way and generally at the expense of introducing more and more parameters. The details of increasingly specic systems can be ever more closely emulated, although this is not altogether surprising as the number of parameters available for ne tuning the model increases. Whether or not more understanding is gained by such phenomenological processes is an open question, but a comprehensive study of dierent models can be enlightening.
Models have been developed which focus on many particular aspects of the dynamics and evolu- tion of interacting populations which do not even feature in the basic systems. A good example is the question of sexual reproduction. The models discussed so far have implicitly assumed asexual reproduction (i.e. all individuals are able to reproduce independently), but many organisms (and
most higher animals) reproduce sexually. At rst sight, sexual reproduction, which is an ancient be- haviour, appears to be an extremely wasteful process: With half the population comprising `useless' males, `rival' asexual individuals should have a two-fold reproductive advantage. In evolutionary terms, this is a huge advantage, and one would expect the sexual species to be doomed. Models which explicitly represent sexual populations have suggested a possible solution to this problem: sex is a mechanism for escaping excessive parasite burdens (Hamilton, Axelrod and Tanese, 1990). Par- asitism is a ubiquitous life-style, and virtually all organisms suer it to some extent. If susceptibility to a parasite is in any way, or in any part, genetically determined (as is possible), then genetically identical asexual populations could be much more vulnerable than the variable individuals in a sex- ual population. We can also ask why there are only two sexes, or what are the benets of being hermaphrodite. A key feature of this explanation is the presence of genetic variability within the population, which sex can mix and recombine. At a more fundamental level, models with explicit genetic structure are widely studied to understand the consequences of dynamics at the level of the gene.