WLF 448: Fish & Wildlife Population Ecology 2011

Lab 5: Exponential Population Growth

I. Introduction

        Is our population of interest increasing or decreasing and how fast?

A. Characteristics of a good model

1. general
2. realistic
3. precise / accurate

B. Deterministic vs. stochastic models

·       Deterministic = No randomness in the model/data… future is completely predictable

·       Stochastic = Random fluctuations in the model/data… future is probabilistic

C. Fundamental population growth parameters

lambda = N_t+1/N_t =  finite rate of increase of the population in one time step (often 1 yr).

o   When:                 0 < lambda <1, population decreases

lambda = 1, population stable

lambda > 1, population increases

r = ln[lambda] = ln[N_t+1/N_t] = instantaneous rate of increase;

lambda = e^r , where e = 2.71828 (= natural log or log to the base e).  

o   When:                 r < 0, population decreases

r = 0, population stable

r > 0, population increases

II. Exponential or Geometric Population Growth Models

A. Assumptions

1.     Population growth or decline follows an exponential curve.

2.     Growth rates are constant (for deterministic growth).

3.     Mean growth rates are constant (for stochastic growth with environmental fluctuations).

·       Situations in nature where exponential growth may occur over appreciable periods of time:

1.     Where a species has naturally colonized or has been introduced by humans into a new and acceptable geographic area.

2.     Where a species has been greatly depressed by human activities and such activities cease.

3.     When there is a constant environmental stress that causes survival and reproduction to consistently be below sustainable levels.

B.  Deterministic model:

For a population with discrete breeding periods, the population size at time t (denoted N_t ) is calculated as:

N_t = N₀ x lambda^t

where N₀ = initial population size

 t = time steps into the future (often expressed as years).

In some cases, it will be easier to work with the equation for exponential growth if we take the natural logarithm of both sides of the equation…

ln[N_t] = ln[N₀] + ln[lambda] x t

If we set ln[lambda] = r ,  then this is an equation describing a line with y-intercept at ln[N₀] , and slope = r

ln[N_t] = ln[N₀] + rt

C.  Stochastic models:

1.     Stochasticity due to Observation error

·        Deviations from deterministic growth model N_t = N₀ x lambda^t  are due to errors in the estimates of abundance (i.e., population growth is deterministic)

·        Our observed population sizes (on a log-scale ) are:

ln[N_t] = ln[N₀] + rt + E

ln[N₀] + rt  is the part for deterministic growth (the actual population abundances)and E is the observation error.  E is a normal random deviate with mean = 0 and standard deviation = tau

E ~ normal(0, tau²)

·       So observed deviations from deterministic growth are random and the amount that it differs from deterministic is controlled by tau

2.     Stochasticity due to Process noise  

·        Deviations from deterministic growth model N_t = N₀ x lambda^t  due to the fact that the growth rate during each time step is stochastic (i.e., population growth itself is stochastic).  For example, variation in environmental conditions could result in 'good' and 'bad' years.

·        So, annual instantaneous growth rates follow:  ln[N_t+1/N_t] =  r + F, where F is a normal random deviate with mean = 0 and standard deviation = sigma.

F ~ normal(0, sigma²)

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 + F.

·        Another way to put this is that our observed instantaneous growth rates are normally distributed with a mean = r and a standard deviation = sigma.  Thus, the average growth rate is constant (i.e., r), but each annual growth can deviate from this average.

ln[N_t+1/N_t] ~ normal(r, sigma²)

D.  Estimating Parameters with Observed Data:

1.     Stochasticity due to Observation error

Remember,

ln[N_t] = ln[N₀] + rt + E

is the model for how our population grows.  Or equivalently,

ln[N_t] ~ normal(ln[N₀] + rt, tau²)

In fact, this is the equation for a linear model whose parameters can be estimated using simple linear regression.

The response variable (i.e., Y) is ln[N_t], the predictor variable (i.e., X) is t.  We can do this in Excel by plotting the natural logarithm of our observed population sizes ln[n_t] against time (t).  Then, we “fit” a linear trend to this plot and display the equation.

The Y-intercept is our estimate of ln[n_t], the slope is our estimate of r.

Exponentiate our estimate of ln[n_t], Exp(ln[n_t]), and this gives us our estimated initial abundance at time = 0.

Exponentiate our estimate of r,  Exp(r), and this gives us our estimate of lambda.

2.     Stochasticity due to Process noise

Remember,

ln[N_t+1/N_t] ~ normal(r, sigma²)

is the model for how our population grows.  Thus, we can estimate r by simply taking the average of our observed instantaneous growth rates,

average (ln[n_t+1/n_t]) = estimate of r

Revised: 22 September 2011