# Introduction

## The Model

The LPA competition model is:

The lower case letters refer to the total number of each lifestage for one species and the uppercase ones for the other.

These equations define what is known as a discrete dynamical system. They model the change in a system over time by taking current values and returning values corresponding to the next time step. In other words they do not give the continuous state of a system but rather data in intervals, two weeks in our case. To look at the behavior of the model system it is easiest to look at a graphical representation of it. Unfortunately there are six different values in the system, so we cannot visualize everything the model reveals. Instead we looked at two dimensional plots generated by adding the values of larva, pupa, and adults of one species and plotting this number versus the sum of larva, pupa, and adults of the other.

A careful examination of the equations will reveal eighteen different state variables (constants). They define how the system will behave and correspond to natural actions like birth rates, death rates and competition. For a more specific breakdown go here.

Since there are so many different state variables, one would expect many different behaviors of the system. This is in fact the case and by changing the state variables we can see many interesting cases. For instance in situations with high death rates trajectories in the system can behave chaotically! Look:

There are other sets of state variables that lead to chaotic attractors. In fact a previous semesters research project of mine looked at the fractal dimension of some chaotic attractors. Check it out here. Also, if you wish to play with individual trajectories yourself, and have Matlab, then use lpaplot.

Previous work in ecology has looked at the possible outcomes of competition of two species, ones that share resources. Lotka-Voltare theory predicts competitive exclusion, extinction, of a species if the level of competition is high between two species, like those of our flour beetles. Dr. Thomas Park and P.H. Leslie performed experiments with flour beetles in the 1960's and used their data to back up Lotka-Voltare. However, there were some unexplained trials that resulted in coexistence of the species. Using the LPA competition model Dr. Jeff Edmunds found a state variable collection were competition was high and a stable attractor in the coexistence region existed.

In the example below you can see three different trajectories. Two lead to dominance of one species, extinction of the other, while the third shows the trajectory settling in on a steady state two cycle attractor.

It is with this set of state variables that we worked this semester:

The model above returns real number values for beetles but in the reality beetles come in integer values. Therefore, we placed the system on an integer lattice and examined trajectory behavior in this context. We chose to examine the behavior of the system with this coexistence phenomenon, on the lattice, to see what ecological and mathematical conclusions we could come to in regards to the effects of volume, rounding, and stochasticity. Read on.

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