0.6 Second order model

Second-order or two-state variable systems

In the last few sections, we discussed first-order models of various systems and studied the types of interactions that could be modeled andthe nature of the solutions of these models. Of the several indicated generalizations that could be made, thissection will consider adding another state variable, so that the effects of two interacting variables can be used and studied.This will greatly increase the class of systems we can model and the class of solutions that result.In addition, a very interesting set of classical problems fall into this class with interesting solutions and interpretations.

To illustrate the general problem, consider a system that contains populations of two different types with distinctly different characteristics.Assume these two populations have a strong effect on each other, as well as being influenced differently by their environment, so that modelingthem by a single total population would not yield useful results. We must, therefore, have two separate state variables to describe thesystems, and this could perhaps be done in the following way.

Here the rate of change of population $p_1$ is assumed to depend on both the populations $p_1$ and $p_2$ ; and likewise, the rate of change of $p_2$ is assumed to depend on $p_1$ and $p_2$ , but in perhaps a different way.

Many types of interactions could be considered. It might be that $p_1$ and $p_2$ compete for the same source of food or resources;it might be that $p_1$ is a prey of the predator $p_2$ ; or it could be that they both contribute to the welfare of the other.These assumptions would be implemented in the choice of $\underline{f}$ and $\underline{g}$ to describe the particular case. The best known classical models of these types were proposed byLotka (1925) andVolterra (1926). Later,Gause (1934) did further experimental and interpretative work.Most of this type of work was done in population ecology. [link] , [link] .

• A The Simple Lotka-Volterra Competition Model

Consider the particular for for the two-variable model to be

This might be a simple model of two competing populations, where $a$ and $c$ are the net rate of increase that would occur if the other population did not exist.The coefficients $b$ and $d$ model the negative effects of interaction on the rates of change as a measure of how often oneencounters the other.

To simplify the mathematics, a change of variables will be made. Consider the rearrangement of [link] into

Now let $x=\left(,\frac{d}{c},\right)p_1$ and $y=\left(,\frac{b}{a},\right)p_2$ then, [link] becomes

Note that $x$ and $y$ are related to $p_1$ and $p_2$ by simple constantmultipliers or scale factors, and therefore, the nature of the solution of [link] is the same as [link] , but now there are only two parameters, $a$ and $c$ , to consider. In fact, by allowing a change of scale of the time variable, it ispossible to reduce the number of parameters to one, but we will not do that.The problem of solving the coupled equation of [link] or, more generally, of [link] can be approached three ways. In some cases, an analytical equation for the solution can be found.This is always true if the equations are linear, but unfortunately, almost never true if they are nonlinear.Another approach was the phase plane where one solution is plotted as a function of the other, with time as an implicit variable.Very important characteristics of the solution can often be determined by phase plane methods without actually finding the solution.Finally, the equations can be numerically solved by simulation on a digital computer using Euler's method or some other more efficientalgorithm.