Project 24 studies a dynamical system; this is really a more general way of looking at a set of differential equations. We are already used to two simple kinds of differential equations, initial value problems and two point boundary value problems. We also looked briefly at the sensitivity of the solution of a system of differential equations with respect to a parameter. However, in all these cases, we were generally concerned with getting a solution.
To talk about a set of differential equations as a dynamical system, we are stepping back and taking a longer view. It's a little like asking about the climate, rather than the weather. We don't want to know exactly what tomorrow's weather will be like; rather, what are the ranges of weather we are likely to see, and are there any unusual patterns of behavior?
In a similar way, if we look closely, not at the solutions, but at the differential equations that we solve to get the solutions, we can ask what effect these equations have, what kind of solutions they allow, and whether the differential equations suggest that the solutions are sensitive to initial conditions, or can oscillate periodically, or become chaotic.
The particular case study for this project involves the life cycle of flour beetles, which go through the stages of larva, pupa, and adult. As in the epidemic model of project 6, the three populations affect each other, in a way that can be described by differential equations. Interestingly, the beetles can at times be cannibalistic!
By looking at different initial conditions, we discover the existence of equilibrium solutions; we can ask whether these solutions are stable, that is, likely to persist despite small perturbations. We will also see strange cases of bifurcation. For certain cases, the system exhibits the classic features of chaos.
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