asbio (version 1.9-2)

anm.LV: 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()

Value

The functions return descriptive animated plots

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.

Author

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

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.

References

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

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

Examples

if (FALSE) {

#---------------------- 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))}