In this case, we have a non-trivial equilibrium solution, i. This equilibrium is reached always, regardless of the initial conditions. The logistic growth model is appropriate for modeling, for example, the growth of bacteria inside a Petri dish under controlled conditions, where the resources are clearly limited and there are no seasonal variations of, say, temperature or humidity.
A general modeling framework for describing spatially structured population dynamics
Its main flaw is that the model is completely blind to an interaction between different species. A straightforward way for obtaining a more realistic model is adding a periodical dependence of the carrying capacity with time. This is a very simple example of yearly variations due to the different seasons:. The explicit presence of time in the differential equation turns it into a non-autonomous one. This makes it harder to solve analytically.
Here we plot a family of solutions obtained numerically:. This is just a model, but it gives us the flavor of the type of variations we can introduce in order to get better models. The first model with interaction between species is due to Alfred J. Lotka and Vito Volterra. Lotka proposed it while studying autocatalytic chemical reactions.
Volterra discovered them independently while studying the evolution of the fisheries of the Adriatic Sea. It had been observed that the populations of the fisheries oscillated with non-yearly frequencies, and it was thought to be caused by the fishing activity. Surprisingly, when World War I stopped almost all fishing activity in the Adriatic, the oscillations prevailed.
The idea is that the population of predators is a death factor for the preys, and the population of preys tends to rise the rate of birth of the predators. The predator — prey equations look like:.
This is a non-linear system. It can be proven that it has periodical solutions, but cannot be solved analytically. Here we plot a possible numerical solution:. It is interesting to note that the system is intrinsically periodic, even we have not introduced any seasonal term. Currently, most of the models are refinements and generalizations of the Volterra-Lotka one. Being even the simpler models non-linear, almost all the models have to be solved numerically.
The field is taking advantage of the rising power of computation and the developments of the non-linear systems theory. The rise of network theory allowed gaining insight into interaction models with more than just two species, treating each species as a node, and each interaction coefficient as a directed edge. This model is known as food web.
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Some of the newest models are taking into account the discrete nature of the specimens, using stochastical simulation to model discrete deaths and births, and even mutations treated as changes in the coefficients, in the simplest cases. There have been also applications of game theory to this field, usually through more complex simulations taking into account space and not only time. The math of sex and hunger. Early developments The first reference of a problem on population dynamics is due to Leonardo de Pisa, also known as Fibonacci, who studied the growth and reproduction of rabbits in the early years of the XIII century.
Population dynamics - Wikipedia
He proposed a time-discrete model like this: Step 0: we start with a young immature couple of rabbits Step 1: the couple of rabbits is now mature. Step 2: the couple of rabbits give birth to a second immature couple of rabbits. See figure below: Figure 1.
Credit: The Artblot The number of alive couples at each step corresponds to the well known Fibonacci series, which is formed by summing the previous two elements in the list. That is: being r the proportionality constant, and N the number of specimens.
Mathematical Population Dynamics and Epidemiology in Temporal and Spatio-Temporal Domains
This improved model is known as logistic model: where K is known as carrying capacity for reasons that will become obvious below. The family of solutions looks like: Figure 2. Logistic growth with seasonal variations A straightforward way for obtaining a more realistic model is adding a periodical dependence of the carrying capacity with time. This is a very simple example of yearly variations due to the different seasons: The explicit presence of time in the differential equation turns it into a non-autonomous one.
Research into the dynamics of ecological populations allows us to understand the conservation of endangered species, make predictions about the effects of global climate change and understand the population fluctuations observed in the natural world.
Work on infectious diseases clearly has important applications to public-health, allowing us to predict the spread of an epidemic such as Foot-and-Mouth or SARS virus and determine the effect of control measures. Throughout, use will be made of examples in the recent literature, with a strong bias towards read-world problems. Special attention will be given to the applied use of the models developed and the necessity of good quality biological data and understanding. Much of this course will be based on research papers and comprehensive references will be given throughout the course.
Four useful books are:.
Modeling cell population dynamics
Anderson and R. ISBN X.
Ellner and J. May and A.
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