An obvious problem with the exponential growth model is that the per-capita rate of increase is assumed to be constant whereas, in reality, it is usually affected by environmental variations as well as by the population itself. Ecological theory leads us to propose that the rate of increase of the average organism in a population can be affected by density dependent factors that respond to the density of the population and feed back to affect the individual at a later date through its per-capita rate of increase, R , and density independent factors that act on R independently of population density and can be considered, for now, as random variables. We can include these concepts in our model by allowing the per-capita rate of change to be a function of its own density plus random environmental influences
where f is a density dependent regulation function, or R -function, d is the average delay in the response of the density dependent forces, and V is a random variable. Density independent forcing variables are assumed to act on the parameters of the model.
Delays in the density dependent feedback response have important effects on the dynamics of populations. For example, long time lags can produce cycles of abundance. Time delays are characteristic of particular kinds of ecological processes. For example, when d = 1 the density dependent factors react to population density, and feed back to affect R , within 1 time step (e.g., 1 generation or 1 year). Rapid density dependent feedback such as this occurs when the survival and reproduction of the members of the population depend on current population density. This is what we have called the third principle of population dynamics. Biological mechanisms that can give rise to rapid feedback are competition for food or space, the behavioral responses of predators or parasitoids (i.e., density dependent aggregation and switching or, conversely, competition for enemy-free-space), and pathogen epizootics. Longer time-delays (d > 1) can be created when some component of the environment changes quantitatively (numerically) in response to changes in the density of the population; e.g., predator reproduction, food depletion, accumulation of pollutants, and so on. What we have called the fourth principle of population dynamics. These quantitative alterations of the environment are often fed back to affect the per-capita rate of change of future generations. Thus, long time-delays often imply that density dependence is due to trophic interactions, as would be caused by numerical interactions with predators, parasites, or food supplies.
Although there are several different models for the population regulation function, P1b only utilizes the discrete nonlinear logistic model (Berryman 1991, 1992, 1994):
A = the maximum per-capita rate of increase, or the intercept of the R -function with the y-axis; i.e., the value of R at N = 0. C = the coefficient of intraspecific competition, or the marginal effect of adding one individual on the per-capita rate of change of its cohorts. Q = the coefficient of curvature: When Q > 1 the R-function bends downwards at high density and has a convex form, and when Q < 1 it has a concave shape. d = the time delay in the dominant density dependent feedback mechanism.
The logistic model can also be written
where K = A / C is the high-density root or carrying capacity of the environment (note that K intercepts the R = 0 line (x-axis) at a relatively large value of N). K is the maximum population density that can be sustained indefinitely in a given environment.
Exogenous density independent factors, which are not involved in density dependent feedback, are assumed to act in three major ways:
1. They can cause random variation from the expected density dependent feedback function. Random effects are usually the result of changes from the average conditions experienced by the organism, like unusually hot or cold weather, and are modeled as normal deviations from the expected R-function. Hence, the logistic R-function becomes
where V is a random normal variate with mean zero and standard deviation s. Environmental fluctuations are simulated by adding random deviates from a normal distribution with given standard deviation to the calculated values of R.
2. They can cause trends in the values of parameters A or K. Environmental forcing of this kind is usually due to environmental factors that change continuously with time, like global warming, and is modeled by altering the parameters as a linear function of time; i.e.,
where X(t) is the value of parameter A or K at time t, X(0) is the original value of the parameter, and b is the slope of the regression of X(t) on t.
3. They can cause steps or discontinuities in the values of the parameters A, K or d. Environmental forcing of this kind is usually the result of sudden changes in the environment, like volcanic eruptions or pesticide sprays, and is modeled as a step function
where X(new) is the new value of the parameter and TSTEP is the time at which the parameter changes its value.
The R-functions described above can have but one equilibrium. Because the second principle of population dynamics, or Allee's principle, makes it possible for populations to have two potentially stable equilibria, P1b allows you to describe data with two R-functions separated by a line, the separatrix, which defines the basins of attraction of the two R-functions
Cl = the coefficient of competition for the lower R-function, dl = the time lag of the lower R-function, Ql = the coefficient of curvature of the lower R-function, U = the unstable escape threshold. The separatrix is defined by the set of R-values that, at given population densities, creates the escape density, U; i.e.,
The models described above have the following dynamical properties:
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