Ergebnis für URL: http://alexei.nfshost.com/PopEcol/lec5/logist.htmlLogistic 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:
[eq4.gif]
[glogist1.gif] At low densities (N < < K), 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:
[eq5.gif]
which has the following solution:
[eq6.gif]
Three possible model outcomes
[glogist2.gif]
1. Population increases and reaches a plateau (No < K). This is the logistic
curve.
2. Population decreases and reaches a plateau (No > K)
3. 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.
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[4]Alexei Sharov 2/03/97
References
1. http://alexei.nfshost.com/PopEcol/lec5/exp.html
2. http://alexei.nfshost.com/PopEcol/lec5/explog.html
3. http://alexei.nfshost.com/PopEcol/lec5/discrete.html
4. http://alexei.nfshost.com/~sharov/alexei.html
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Errormessages are in German, sorry ;-)