LIFESCI 30A Lecture Notes - Lecture 5: Logistic Function, Mathematical And Theoretical Biology, Ferris Wheel
Document Summary
Our very first mathematical model was the following simple model of the tub- faucet-drain system: Two parameters f = 10 l/min and k = 0. 2 1/min. Now we are going to introduce a new feature: interaction of state variables. We will consider two fundamental models with interaction: Our system is a population of animals (rabbits) that are competing for the same resources (lettuce). X(t) = number of rabbits at time t. We will tally up the inflows and the outflows : The idea is that the more rabbits there are, the more likely they are to steal resources from one another. Prob(two rabbits go for the same bit of lettuce l) = prob (a rabbit -> l) * prob (a rabbit -> l) This is the logistic equation from mathematical biology. The key new feature is the interaction term x2. We get another form of this equation by factoring out bx(t): Thus, we can rewrite the logistic equation as:
