Exponential and Logistic Growth
Exponential and Logistic
Growth
We can describe the sigmoid growth curve (see Figure 40-4) by a simple model called the logistic equation. The slope at any point on the growth curve is the growth rate, how rapidly the population size is changing with time. If N represents the number of organisms and t the time, we can, in the language of calculus, express growth as an instantaneous rate:
Under these ideal conditions growth is expressed by the symbol r, which is defined as the intrinsic rate of population growth per capita. The index r is actually the difference between the birth rate and death rate per individual in the population at any instant. The growth rate of the population as a whole is then:
This expression describes the rapid, exponential growth illustrated by the early upward-curving portion of the sigmoid growth curve (see Figure 40-4).
Growth rate for populations in the real world slows as the upper limit is approached, and eventually stops altogether. At this point N has reached its maximum density because the space being studied has become “saturated” with animals. This limit is called the carrying capacity of the environment and is expressed by the symbol K. The sigmoid population growth curve can now be described by the logistic equation, which is written as follows:
This equation states that the rate of increase per unit of time (dN/dt = rate of growth per capita (r) × population size (N) × unutilized freedom for growth ([K - N]/K). One can see from the equation that when the population approaches the carrying capacity, K - N approaches 0, dN/dt also approaches 0, and the curve will flatten.
Populations occasionally overshoot the carrying capacity of the environment so that N exceeds K. The population then exhausts some resource (usually food or shelter). The rate of growth, dN/dt, then becomes negative and the population must decline.
We can describe the sigmoid growth curve (see Figure 40-4) by a simple model called the logistic equation. The slope at any point on the growth curve is the growth rate, how rapidly the population size is changing with time. If N represents the number of organisms and t the time, we can, in the language of calculus, express growth as an instantaneous rate:
-
dN/dt = the rate of change in the
number of organisms per time at
a particular instant in time.
Under these ideal conditions growth is expressed by the symbol r, which is defined as the intrinsic rate of population growth per capita. The index r is actually the difference between the birth rate and death rate per individual in the population at any instant. The growth rate of the population as a whole is then:
| dN/dt = rN |
This expression describes the rapid, exponential growth illustrated by the early upward-curving portion of the sigmoid growth curve (see Figure 40-4).
Growth rate for populations in the real world slows as the upper limit is approached, and eventually stops altogether. At this point N has reached its maximum density because the space being studied has become “saturated” with animals. This limit is called the carrying capacity of the environment and is expressed by the symbol K. The sigmoid population growth curve can now be described by the logistic equation, which is written as follows:
| dN/dt = rN([K - N]/K) |
This equation states that the rate of increase per unit of time (dN/dt = rate of growth per capita (r) × population size (N) × unutilized freedom for growth ([K - N]/K). One can see from the equation that when the population approaches the carrying capacity, K - N approaches 0, dN/dt also approaches 0, and the curve will flatten.
Populations occasionally overshoot the carrying capacity of the environment so that N exceeds K. The population then exhausts some resource (usually food or shelter). The rate of growth, dN/dt, then becomes negative and the population must decline.
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