Wildlife Ecology

This page calculates population growth and maximum sustainable yield. The graphing requires the CANVAS HTML tag, so will not work in IE6.

Input
gest=4 // gestation period in years litter=1 // no. of surviving offspring per birth fem=0.5 // % of population that is female fert=0.75 // % of females that are fertile life=35 // average longevity in years K=10000 // carrying capacity N0=100 // starting population dur=200 // graphing duration in years // lambda=1+litter/gest*fem*fert - 1/life output+="Lambda: "+lambda.toFixed(2)+"\n" r=log(lambda) output+="Malthusian r: "+r.toFixed(3)+"\n" dNmax=r*K/4; K2=K/2 output+="Maximum sustainable yield: "+dNmax.toFixed(0)+" at N="+K2+"\n" t50=-1/r*log(1/(K/N0-1)) output+="Time to 1/2 K: "+t50.toFixed(1)+" years\n" dt=dur/600 output+="t\tdN/dt\tN\n" N=N0 canvas.clearRect(0,0,601,401) canvas.fillStyle = "rgb(0,0,0)" canvas.strokeRect(1,1,599,399) canvas.fillRect(0,200,600,0.5) x=t50/dur*600 canvas.fillRect(x,0,0.5,400) for(t=0;t<=dur;t+=dt) { oldN=N N=K/(1 + (1/N0*K - 1)*exp(-r*t)) avgN=(N+oldN)/2 dNdt=r*avgN*(K-avgN)/K output+=t.toFixed(2)+"\t"+dNdt.toFixed(0)+"\t"+N.toFixed(0)+"\n" x=t/dur*600; y=N/K*399 canvas.fillStyle = "rgb(0,0,255)" canvas.fillRect(x,400-y,2,2) y=dNdt/dNmax*300 canvas.fillStyle = "rgb(255,0,0)" canvas.fillRect(x,400-y,2,2) }

Output

Canvas

This page takes adjustable parameters and calculates the rate of population growth and the maximum sustainable yield.

The input parameters can be altered in the first text box.

gest: the gestation period, or time between births, in years
litter: the number of offspring per birth that survive
fem: the percentage of the population that is female
life: the average longevity in years
K: the carrying capacity, or maximum population level that resources can support
N0: the initial population
dur: the length of time that will be graphed

By pressing the Go! button, the Javascript calculates lambda, the maximum yearly increase (or decrease) in the population via the following equation:

lambda = 1 + litter/gest*fem*fert - 1/life

This result along with several other values is displayed in the output text box. Values of lambda greater than one indicate that the population is increasing, while values less than one indicate the population is decreasing.

A related value, r, which is a rate constant for the maximum yearly increase is derived from lambda.

r=log(lambda)

Values of r greater than zero indicate that the population is increasing, while values less than zero indicate the population is decreasing. This parameter is sometimes referred to as the Malthusian rate due to its relation to exponential population growth.

If you like, you can substitute values for lambda or r directly, rather than have the Javascript calculate them.

The rate of population growth is given by:

dN/dt = rN(K-N)/K

The rate is low when N is small, increases to a maximum when N is one-half the carrying capacity, K, and decreases to zero as N approaches K. In the graph box the rate of growth is shown by the red line. The maximum rate is given by:

dN/dt max = rK/4

This rate is the maximum number of animals that can be harvested per year in a sustainable fashion. This maximum rate occurs at the following time:

t(50%) = -1/r*log(1/(K/N0-1))

The population at any time is given by the following logistic equation.

N=K/(1 + (1/N0*K - 1)*exp(-r*t))

In the graph the population is shown by the blue line. A horizontal line shows where N equals one-half of K, and the vertical line shows where the maximum rate of growth occurs.

The values of time, N and dN/dt are provided in tab-delimited format in the output text box. You may copy and paste these values directly into a spreadsheet.

The default parameters represent those typical of a large animal, such as a whale, with a long gestation period, a small litter and a long life-time.

You may save changes you have made to the parameters to a cookie by pressing the Save button. Likewise, on the same computer you can come back later and Load the saved parameters from the cookie.

Jeffrey Clymer • June 5, 2009 • Homepage