anm.LV: Animated depictions of Lotka-Volterra competition and...

In asbio: A Collection of Statistical Tools for Biologists

Animated depictions of Lotka-Volterra competition and exploitation models

Description

Creates animated plots of two famous abundance models from ecology; the Lotka-Volterra competition and exploitation models

Usage

anm.LVcomp(n1, n2, r1, r2, K1, K2, a2.1, a1.2, time = seq(0, 200), ylab = 
"Abundance", xlab = "Time", interval = 0.1, ...)

anm.LVexp(nh, np, rh, con, p, d.p, time = seq(0, 200), ylab = "Abundance", 
xlab = "Time", interval = 0.1, circle = FALSE, ...)

anm.LVc.tck()

anm.LVe.tck()

Arguments

n1	Initial abundance values for species one. To be used in the competition function anm.LVcomp, i.e., N_1 in the competition equations below.
n2	Initial abundance values for species two in the competition function, i.e., N_2 in the competition equations below.
r1	Maximum intrinsic rate of increase for species one, i.e., r_{max1}.
r2	Maximum intrinsic rate of increase for species two in the competition model anm.LVcomp, i.e., r_{max2}.
K1	Carrying capacity for species one, i.e., K_1.
K2	Carrying capacity for species two, i.e., K_2.
a2.1	The interspecific effect of species one on species two, i.e., the term \alpha_{21}.
a1.2	The interspecific effect of species two on species one, i.e., the term \alpha_{12}.
nh	Initial abundance values for the host (prey) species. To be used in the the exploitation model anm.LVexp, i.e., the term N_h at t = 1.
np	Initial abundance values for the predator species in the the exploitation model, i.e., the term N_p at t = 1.
rh	The intrinsic rate of increase for the host (prey) species, i.e., the term r_h.
con	The conversion rate of prey to predator, i.e., the term c.
p	The predation rate, i.e., the term p.
d.p	The death rate of predators, i.e., the term d_p.
time	A time sequence for which competition or exploitation is to be evaluated.
ylab	Y-axis label.
xlab	X-axis label.
interval	Animation speed per frame (in seconds).
circle	Logical, if TRUE a circular representation of the relation of prey and predator numbers is drawn.
...	Additional arguments from plot.

Details

The Lotka-Volterra competition and exploitation models require simultaneous solutions for two differential equations. These are solved using the function rk4 from odesolve.

The interspecific competition model is based on:

\frac{dN_1}{dt}=r_{max1}N_1\frac{K_1-N_1-\alpha_{12}}{K_1},

\frac{dN_2}{dt}=r_{max2}N_2\frac{K_2-N_2-\alpha_{21}}{K_2},

where N_1 is the number of individuals from species one, K_1 is the carrying capacity for species one, r_{max1} is the maximum intrinsic rate of increase of species one, and \alpha_{12} is the interspecific competitive effect of species two on species one.

The exploitation model is based on:

\frac{dN_h}{dt} = r_hN_h-pN_hN_p,

\frac{dN_p}{dt} = cpN_hN_p-d_pN_p,

where N_h is the number of individuals from the host (prey) species, N_p is the number of individuals from the predator species, r_h is the intrinsic rate of increase for the host (prey) species, p is the rate of predation, c is a conversion factor which describes the rate at which prey are converted to new predators, and d_p is the death rate of the predators.

The term r_hN_h describes exponential growth for the host (prey) species. This will be opposed by deaths due to predation, i.e. the term pN_hN_p. The term cpN_hN_p is the rate at which predators destroy prey. This in turn will be opposed by d_pN_p, i.e. predator deaths. Loading package tcltk allows one to run the GUIs in anm.LVe.tck and anm.LVc.tck.

Value

The functions return descriptive animated plots

Author(s)

Ken Aho, based on a concept elucidated by M. Crawley

References

Molles, M. C. (2010) Ecology, Concepts and Applications, 5th edition. McGraw Hill.

Crawley, M. J. (2007) The R Book. Wiley

Examples

## Not run: 

#---------------------- Competition ---------------------#
##Species 2 drives species 1 to extinction
anm.LVcomp(n1=150,n2=50,r1=.7,r2=.8,K1=200,K2=1000,a2.1=.5,a1.2=.7,time=seq(0,200))
##Species coexist with numbers below carrying capacities
anm.LVcomp(n1=150,n2=50,r1=.7,r2=.8,K1=750,K2=1000,a2.1=.5,a1.2=.7,time=seq(0,200))

#----------------------Exploitation----------------------#
#Fast cycles
anm.LVexp(nh=300,np=50,rh=.7,con=.4,p=.006,d.p=.2,time=seq(0,200))
## End(Not run)

asbio documentation built on May 29, 2024, 5:57 a.m.