Plant Herbivore System Model the dynamics involved in intro

Plant - Herbivore System

Model the dynamics involved in introducing a pollution of 100 deer into an area consisting of a patchwork of grassland and forest. By Dynamics we mean we want to predict the type of behavior seen like growing, declining, oscilatting, chaotic and we are not intrested in predicting actual numbers of animals in a given year. Rather than count individual plants, we measure the plant density in appropriate density units. In the absence of deer plants observe a logistic type growth with a growth rate of 0.8 when the plant density is low and growth rate near 00?when the plant density is near 3000 desity units. The deer herd eats the plants of course and when the vegetation is burned out or grazed flat, the herd will decline at a rate of 1.1(that is the number of deer lost in a year is 1.1 times the number of deer which is impossible but it only means that the herd would be gone in less than a year with no food.) when food is present this decline is offset by a growth term which has a rate 0 when plant density is 0,1.5 when plant desity is 3000 and is roughly linear assume it is linear in between.(this is just the growth rate not the actual number of deer.)Similarly when deer are present the plant density is diminished. Each year can eat as much as 1.2 density units per year when food is plentiful (plant desnity is 3000), but as food becomes scarcer the deer will eat less with no food eaten when there is no food available. again assume a linear realtionship between eating no food when the plant density is zero and 1.2 density units when the density is 3000.

1) Construct a simulation model and run it for 30 years under the follwoing tree conditions a) Discrete time steps of one year , b) discrete time steps of one day c) continious time step. For each of this cases make a graph with deer and plant density plotted as a fuction of time on the same axis, also make a phase plane plot for each case deer on one axis plant on the other. Write down the existence and stability of fixed points and the differences between the various cases which would be most appropriate ?

2) The situation above is a modification of a model by caughley which is indentical except that the interactions terms for the deer and plants do not vary linearly. His model has decrease in plant density due to deer equal to 1.2 deer population (1-exp(-.001 * plant density)) and the increse in deer due to plant as 1.5 deer population*(1-exp(-0.001*plant density)). Replace the plant-deer interaction terms with caughley\'s terms and repeat the work of Part 1)

3) Compare the models 1) and 2) plot the 1-exp(-0.001x)) from 0 to 5000 and explaning what this term contributes to the second model over the line used in the first model.Which model does the best job of explaining the obeserved behaviour.

Solution

given Number of deers = 100

their growth rate when plant density is low = 0.8

plant density= 3000 density units

when plant density is 3000 the food consumed by deers is 1.2 density units and when plant density is zero there is no food consumed by deers.