Under simplified conditions, such as a constant environment (and with no migration), it can be shown that change in population size (N) through time (t) will depend on the difference between individual birth rate (b_{0}) and death rate (d_{0}), and given by:
where:
The difference between birth and death rates (b_{0} - d_{0}) is also called r, the intrinsic rate of natural increase, or the Malthusian parameter. It is the theoretical maximum number of individuals added to the population per individual per time. By solving the differential equation 1 we get a formula to estimate a population size at any time:
where e = 2.718... (base of natural logs).
This equation shows us that if birth and death rates are constant, population size will grow exponentially. If you transform the equation to natural logarithms (ln), the exponential curve becomes linear, and the slope of that line can be shown to be r:
where ln(e) = 1. The population growth rate, r, is a basic measure in population studies, and it can be used as a basis of comparison for different populations and species.
We need to modify the basic equation 1 so that birth and death rates are no longer constant through time, but decrease and increase respectively as population size increases:
where k_{b} and k_{d} are the density-dependent birth and death rate constants respectively. This equation predicts that a population will stop growing (zero population growth) when birth rate equals death rate, or:
This can be converted into an equation showing the size at which the population reaches a steady state:
The value of N when the population is at steady state is the carrying capacity of the environment, or K. This can be simplified:
since b_{0} - d_{0} = r. If we combine this new form of the carrying capacity equation with equation 5 we get the familiar form of the logistic growth equation:
Mathematically, competition between species is treated as another density-dependent effect on the carrying capacity for each species. In the logistic equation 9, you have already been introduced to the density-dependent birth and intraspecific competition: as the number of individuals in one population increase, birth and death rates change because resources become limiting.
To incorporate the effects of interspecific competition (that is the interaction between individuals of different species) we simply include a competition term in the logistic equation 9. For the purposes of the model, competition now becomes defined as "the amount by which Species 1 lowers the K of Species 2, and vice versa," so that there are two competition coefficients for a two-species system.
Recall the basic logistic growth equation 9. For Species 1, we can incorporate a competition coefficient, , into this:
where is the number of individuals of competing Species 2 which are equivalent to Species 1. Similarly, for Species 2, is the strength of competition it experiences from Species A, incorporated as:
where:
The competition coefficient determines how much of an effect the population of another species will have on the growth rate of a species. For example, if Species 1 has a competition coefficient, = 1.0, then this means that it is experiencing the same intensity of competition from an individual of Species 2, as it is from an individual of its own species. And if Species 2 has a = 2.0, then it is experiencing much higher interspecific competition relative to intraspecific competition. What would be the outcome in a mixed culture under these conditions? Intuitively, it would seem that Species 1 would win since it is experiencing less competition overall. However, the outcome also depends on the relative intensity of intraspecific competition (or the ratio of the Ks). In this case there is not necessarily just one outcome.
There are four possible outcomes of a two species competitive interaction, as follows:
Species 2 will be eliminated when: < K_{1}/K_{2} and > K_{2}/K_{1}
Coexistence of both species when: < K_{1}/K_{2} and < K_{2}/K_{1}
Which species survives depends on the relative sizes of the starting populations
when:
> K_{1}/K_{2} and
> K_{2}/K_{1}
As an example, in Gause's competition experiment with P. caudatum and S. mytilus, he reported the competition coefficients were = 5.5 and = 0.12 and the carrying capacities were K_{1 }= 75 and K_{2} = 15.6, respectively. These coefficients and the carrying capacities suggest that P. caudatum (Species 1) should be eliminated by S. mytilus (Species 2) since:
5.5 > 75 / 15.6 = 4.8 and 0.12 < 15.6 / 75 = 0.21
The simplest model for predation is the Volterra predator-prey equation which has two parts. The change in prey population through time is described by:
dV/dt = rV - (aV)P
and the change in predator population size through time is:
dP/dt = b(aV)P - d_{0}P
where:
The efficiency factor, b, measures how well the predator converts food into offspring. A less efficient predator will expend more of its energy on maintenance than on reproduction compared to an efficient predator. This model is simplistic because it assumes exponential growth of prey, and a predator can eat as many prey as it can catch -- that is, it never gets full!
The basic model gives several important results:
If the predator and prey populations are roughly in balance (populations rise and fall in the short-term, but the long-term average size of the populations does not change) then several more results accrue:
The following model is more realistic because it incorporates logistic growth for the prey populations, and rather than assume that predation rate is simply related to prey density, it incorporates a predator satiation factor. When the prey are very abundant, a predator can no longer continue to increase its feeding rate because it becomes full. The model is:
dV/dt = r [1 - (V/K)]V - c[1 - e^{(-aV/c)}]P
dP/dt = bc [1 - e^{(-aV/c)})P] - d_{0}P
where:
With this model, oscillations are possible, but they are not the only behaviour predicted. We also get stable coexistence.
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