INTRODUCTION

If an organism were able to grow and reproduce in an environment with unlimited resources, an explosive population growth would result. As you know from your textbook, the number of individuals in a population at any given time is equal to the number present at some time in the past, plus the number born between then and now, minus the number that died, plus the number that immigrated, minus the number that emigrated. If for the moment we ignore immigration and emigration and assume births and deaths occur continuously and at constant rates, then this form of explosive increase is called exponential growth. Mathematically, exponential growth can be expressed as

    (1)

Where dN/dT is the rate of change in the size of the population, b is average per capita birth rate, d is average per capita death rate, and N is the number of individuals. The difference between birth rate b and death rate d can be written as r. Substituting r into equation (1) gives

    (2)

This equation states that the rate of change in the size of the population is simply a constant (r) multiplied by the size of the current population.

Let's experiment with our equation for exponential growth. Select Graph, and try entering r = 1.0, r = 0.0, and r = -0.5. Click on New Plot to see a plot for given values of r and N₀ (initial population size). To see the value of N for a given time, move your cursor to the desired location, then click/tap. Click/tap again to clear.

As you plotted different values of r, you saw that the population increases for r > 0, decreases for r < 0, and stays constant for r = 0. If you think back to equation (1) none of this should have been a surprise.

    (1)

    (2)

If r > 0 then the population birth rate is greater than the death rate and the population increases. If births equal deaths, then the population size stays the same. And if the death rate is greater than the birth rate, the population will decrease.

As you probably have guessed, the value of r in a real population is dependent on the organism and environmental conditions. When conditions are optimal for the population, r has its hightest value, called r_max, the intrinsic rate of increase. r_max has a characteristic value for each species.