The Lotka Volterra equation are used to interpret
the population dynamics in which two organisms interact in one of the two ways,
(a) either compete for common resources or (b) associated in a prey-predator
system.
The equation for first type of interaction i.e.
competing for common resources is termed as the Competitive Lotka-Volterra
equations while another type of interaction is described by predator-prey
equation.
1. Competitive
Lotka-Volterra equations
The
Lotka-Volterra equation for competition is based on the logistic equation. This
equation is similar to Predation prey equation of Lotka-Volterra where species
interact with others by one term and to itself by another term but this
equation follows exponential mode rather than logistic model.
Ecologists
used the equation for logistic model is given as:
dN/dt=
rN (1- N/K)
·
N= population size
·
r=growth rate of
population
·
K=carrying capacity
For competition between
two species
In the Lotka-Volterra
equation two additional terms were added to depict the species interactions
between two given population N1 and N2 related logistic
dynamics.
The equation is given
as:
dN1/dt= r1N1 [1-(N1+a12N1/K1)]
for species 1
dN2/dt= r2N2 [1-(N2+a21N2/K2)]
for species 2
We all know that each
organism has its own carrying capacity (K1 and K2 are
different) and growth rate (r1 and r2 are different). As
this equation for population dynamics is associated with interaction
(competition) which are harmful to interesting species, in equation all a
values are positive. In this equation a12
termed as competition coefficient
depicts the competitive effect on
population one (N1) by another population (N2) that’s
why represent as (a12).
Similarly for the equation of population of N2, a21
depicts the competitive effect on population N2 by the population N1
represented as (a21). If a12<1 it means species 1 has
more effect on its own rather than the
effect of species 2 on species 1 i.e. more intense intra-specific
competition.
In short the effected
population comes first to the population affecting it.
Outcomes
Interpretation
1. If
a12=0, this means species 1 follow logistic model of population
dynamics.
2. If
a21=0, this means species 2 follow logistic model of population
dynamics.
3. If a12=1, species 1 and 2
strongly compete with equal magnitude for common resources means intraspecific
competition equals interspecific competition.
4. If
a12= “-” negative, species 2 facilitates resource availability to
species 1.
5. a12/
a21 both negative indicate symbiotic relationships.
6. If
one is zero (0) among two, either a12 or a21 and the
other is negative it indicates commensalism.
7. If
one is positive among two, either a12 or a21 and the
other has no affect it means parasitism.
8. If
both have negative values it indicates competition.
Lotka Volterra Predation equations
The Predator prey
equations given by Lotka-Volterra describe the interaction between prey and
predation as a dynamic biological system.
The population follows
a non-linear, first order differential equation and represented in pair of
equation as
dx/dt = ax – βxy
dy/dt = δxy- γy
where “x” and “y” is
the number of prey and predator population and “dx/dt” or “dy/dt” are the
growth rate of prey and predator population in time “t”. The symbols α,
β,
γ
and δ denotes the real and positive
parameters related to the interaction between prey and predator species.
Assumptions
Lotka-Volterra
predation equation is based on the assumptions:
1. Predator
can eat limitless.
2. Supply
of food resource (i.e. prey) depends on the prey population size.
3. The
rate of change of population directly depends on its size.
4. Environment
is constant, inconsequential genetic adaptations for both species.
5. All
time unlimited food supply for prey.
The equation is continuous and deterministic
indicated continuous overlapping of prey and predator population.
Prey
The
rate of growth for prey population is given by equation:
dx/dt = ax – βxy
The
change in prey population follows exponential growth model represent as “ax” in
the equation unless subjected to predation. The rate of predation is directly
dependent on the rate of prey-predator interaction represented as “βxy”.
“x” and “y” are the population size of prey and predator, if x/y is zero it
clearly indicates that there is no predation.
Predator
The
rate of growth for predator population is given by equation:
dy/dt = δxy – γy
The
change in predator population equals to food supply mediated growth minus death
of predators.
The
growth of predator population is represented as “δxy”.
“δ”
is different from “β” as the rate of predator
population growth is very much different from the rate of predation on prey. “γy”
is the decay rate of predator either due to emigration or mortality. In the
prey absence the predator follow an exponential decay.
The
solution to equation is periodic and yields a simple harmonic motion where prey
population is traced by predation population by 90ᵒ
in cycle.
Figure 27: Harmonic
motion of prey-predator system
Functional and
numerical responses
In
1959, Holling suggested that as the prey density increases it leads to increase
in Predation rates due to two effects:
(a) Functional Response
where the consumption rate of predator increases in presence of high prey
population density.
(b) Numerical Response
in which increase in prey density leads to increase in predator population
density.
a. Functional Response
There are three types of functional response curves
which relate density of prey population to prey consumption by single predator
per unit time.
Type I Response
curve
It’s a rare form of response in nature, for example
in Filter feeders.
It’s an initial exponential relationship between
prey and predator and its consumption by predator till reaches saturation point
where predator cannot eat maximum from that rate.
Type II Response
curve
It’s a common type of response in nature, for
example rodents and weasels. At low population density the rate of consumption
increases at decelerating rate (increases at slow rate).the rate of consumption
is dependent on two factors, (a) the searching or locating a prey and (b)
handling time of prey (capture, kill and eat). At low population density the
searching for prey is important and predator kill prey at constant effort
fashion while at high density of prey searching for prey becomes easy but
handling time is limiting factor and thus the rate of consumption increases
slowly. Subsequently, searching Is not required and rate of consumption levels
off at maximum rate.
Type III
Response curve
Type III response is also common in nature depicting
logistic increase in the rate of consumption on increase in prey density.
Figure 28: Three types
of Functional response curves
Numerical response
curves
Numerical
response means increase in predator density on increase in prey population
density due to direct responses i.e. (a) on increase in prey density the
fecundity rate of predator increases and (b) Aggregation response
a. As
the prey population size increases, the consumption rate of predator increases
leads to high fecundity rate and low mortality rate.
b. Aggregation
response: Predators aggregation in prey hot spots is called aggregation
response. This type of response increase prey-predator system stability.
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