Exponential Model

Exponential model is associated with the name of Thomas Robert Malthus (1766-1834) who first realized that any species can potentially increase in numbers according to a geometric series. For example, if a species has non-overlapping populations (e.g., annual plants), and each organism produces R offspring, then, population numbers N in generations t=0,1,2,... is equal to:

When t is large, then this equation can be approximated by an exponential function:

There are 3 possible model outcomes:

1. Population exponentially declines (r < 0)
2. Population exponentially increases (r > 0)
3. Population does not change (r = 0)

Parameter r is called:

• Malthusian parameter
• Intrinsic rate of increase
• Instantaneous rate of natural increase
• Population growth rate

"Instantaneous rate of natural increase" and "Population growth rate" are generic terms because they do not imply any relationship to population density. It is better to use the term "Intrinsic rate of increase" for parameter r in the logistic model rather than in the exponential model because in the logistic model, r equals to the population growth rate at very low density (no environmental resistance).

Assumptions of Exponential Model:

1. Continuous reproduction (e.g., no seasonality)
2. All organisms are identical (e.g., no age structure)
3. Environment is constant in space and time (e.g., resources are unlimited)

However, exponential model is robust; it gives reasonable precision even if these conditions do not met. Organisms may differ in their age, survival, and mortality. But the population consists of a large number of organisms, and thus their birth and death rates are averaged.

Parameter r in the exponential model can be interpreted as a difference between the birth (reproduction) rate and the death rate:

where b is the birth rate and m is the death rate. Birth rate is the number of offspring organisms produced per one existing organism in the population per unit time. Death rate is the probability of dying per one organism. The rate of population growth (r) is equal to birth rate (b) minus death rate (m).

Applications of the exponential model

• microbiology (growth of bacteria),
• conservation biology (restoration of disturbed populations),
• insect rearing (prediction of yield),
• plant or insect quarantine (population growth of introduced species),
• fishery (prediction of fish dynamics).

Logistic Model

Logistic Model

Logistic model was developed by Belgian mathematician Pierre Verhulst (1838) who suggested that the rate of population increase may be limited, i.e., it may depend on population density:

At low densities (N < < 0), the population growth rate is maximal and equals to ro. Parameter ro can be interpreted as population growth rate in the absence of intra-specific competition.

Population growth rate declines with population numbers, N, and reaches 0 when N = K. Parameter K is the upper limit of population growth and it is called carrying capacity. It is usually interpreted as the amount of resources expressed in the number of organisms that can be supported by these resources. If population numbers exceed K, then population growth rate becomes negative and population numbers decline. The dynamics of the population is described by the differential equation:

which has the following solution:

Three possible model outcomes

Population increases and reaches a plateau (No < K). This is the logistic curve. Population decreases and reaches a plateau (No > K) Population does not change (No = K or No = 0)

Logistic model has two equilibria: N = 0 and N = K. The first equilibrium is unstable because any small deviation from this equilibrium will lead to population growth. The second equilibrium is stable because after small disturbance the population returns to this equilibrium state.

Logistic model combines two ecological processes: reproduction and competition. Both processes depend on population numbers (or density). The rate of both processes corresponds to the mass-action law with coefficients: ro for reproduction and ro/K for competition.

Interpretation of parameters of the logistic model

Parameter ro is relatively easy to interpret: this is the maximum possible rate of population growth which is the net effect of reproduction and mortality (excluding density-dependent mortality). Slowly reproducing organisms (elephants) have low ro and rapidly reproducing organisms (majority of pest insects) have high ro. The problem with the logistic model is that parameter ro controls not only population growth rate, but population decline rate (at N > K) as well. Here biological sense becomes not clear. It is not obvious that organisms with a low reproduction rate should die at the same slow rate. If reproduction is slow and mortality is fast, then the logistic model will not work.

Parameter K has biological meaning for populations with a strong interaction among individuals that controls their reproduction. For example, rodents have social structure that controls reproduction, birds have territoriality, plants compete for space and light. However, parameter K has no clear meaning for organisms whose population dynamics is determined by the balance of reproduction and mortality processes (e.g., most insect populations). In this case the equilibrium population density does not necessary correspond to the amount of resources; thus, the term "carrying capacity" becomes confusing. For example, equilibrium density may depend on mortality caused by natural enemies.