WLF 448: Fish & Wildlife Population Ecology 2010

Lab 6: Population Growth in Limited Environments

I. Introduction to Inverse Density Dependence

        Is there intra-specific competition in our population of interest?

A. Population growth can be similar to the exponential model only under special circumstances and for short periods of time. In most cases:

1.     As N increases, resources become limited and intraspecific competition becomes important.

2.     As N increases, density-dependent factors negatively affect mortality, survival , and/or birth rates, which lead to a decrease in population growth rate.

3.     When population sizes are small, competitive pressures are released and growth rate increases.

4.     Therefore, population growth rates (or average growth rates) are not constant, they depend on population size (i.e, density dependent).

B. Density Dependent Models:

1.     Ricker (logistic) Model:

·        In the beginning, population growth is nearly exponential, with increases close to r_max.

·        There is a constant linear decrease in the instantaneous growth rate (r) as population size increases.

·        Population size plateaus and fluctuates around some mean.

·        The resulting logistic growth curve is S-shaped (sigmoid curve) and has 3 important characteristics:

o   The curve has an upper asymptote called the carrying capacity (K). This is the maximum population size at which the instantaneous growth rate = 0.

o   Deceleration in population growth is smooth as it approaches K; thus, when the curve is cut in half, the upper and lower halves are mirror images. The point where you could cut the curve in half is called the inflection point, and it occurs at K/2.

o   The maximum number of individuals being "added" to the population per unit of time occurs at N = K/2. Theoretically, this point on the growth curve is the population level at which one could manage for Maximum Sustained Yield (MSY).

        ** Remember for stochastic exponential growth with process noise the model was:

ln[N_t+1/N_t] =  r + F, where F ~ normal(0, sigma²)

or equivalently

ln[N_t+1/N_t] ~ normal(r, sigma²)

Under the assumption of the Ricker model (i.e., "there is a constant linear decrease in the growth rate (r) as population size increases"), it is easy to adapt this for density dependence...

ln[N_t+1/N_t] ~ normal(r_max + bN_t, sigma²)

This is the Stochastic Ricker (logistic) Model where r_max and b are estimated parameters, b measures the magnitude of intraspecific competition.  This allows us to predict next year's population size but it is always dependent on the previous year's abundance:

ln[N_t+1] =  ln[N_t] + r_max + bN_t + F. 

·        Note: this equation is the same for a linear model whose parameters can be estimated using linear regression.

·        The response variable (i.e., Y) is ln[N_t+1/N_t], the predictor variable (i.e., X) is N_t.

·        We can do this in Excel by plotting the natural logarithm of our observed instantaneous growth rates, ln[n_t+1/n_t], against observed population sizes n_t.  Then, we “fit” a linear trend to this plot and display the equation.

·        The Y-intercept is our estimate of r_max, the slope is our estimate of b. 

·        If the estimate of b is negative, there is 'negative density dependence' meaning that the growth rate decreases with increasing abundance.

·        Our estimate of carrying capacity K is

K = -r_max/b

2.     Gompertz Model:

·        The Gompertz model is similar to the Ricker except that there is a constant linear decrease in the instantaneous growth rate (r) as the natural logarithm of population size increases

·        This means that there is much more of a density dependent effect at small population sizes but as population size increases, the effect becomes less and less pronounced.

ln[N_t+1/N_t] ~ normal(r_max + b x ln[N_t],  sigma²)

ln[N_t+1] =  ln[N_t] + r_max + b x ln[N_t] + F

** Note:  For the Gompertz model, there is a slightly different interpretation of r_max.

§  Under the Ricker model, r_max was the growth rate when the population size was at its smallest possible value (i.e, 0).

§  For the Gompertz model, r_max is the growth rate when the population size equals 1.

§  Interestingly, under the Gompertz model, the growth rate goes to infinity as the population size goes to 0!

·        We can obtain estimates of r_max and b by regressing ln[N_t+1/N_t] against ln[N_t].  The y-intercept is our estimate of r_max and the slope is our estimate of b.

·        Under the Gompertz model, our estimate of carrying capacity K is

K = Exp[-r_max/b]

** Both stochastic versions of the Ricker and Gompertz models presented here assume stochasticity arises solely from environmental (process) noise.

Revised: 28 September 2010