It is widely recognized among scientists that mathematics is an essential accessory in scientific studies, and that mathematical models are employed to predict future trends and sometimes to infer information about our past. For example, it is possible to use logarithms to estimate roughly how long ago the most recent common ancestor of everyone on Earth today appeared; or we could also use differential equations to predict the infection development spread of a particular disease. This article will discuss the applications of one of the most commonly known and used mathematical models in Biology - dynamical systems - and give some very simplified examples of them.
Dynamical system is commonly defined as an evolution rule which determines how things change or behave over time, and examples of such application is in modelling the drug resistance in malaria, auto-regulation in kidney, anti-coagulation therapy and Ebola outbreaks. All of these involved extremely long and complex modelling and are therefore beyond the scope of this article. However, we can use a very simple example to help ourselves understand the principles of dynamical systems. For a population of rabbits, the area in which they live in might only support a population size of 1500. If the population is small enough, the population size will increase 1.5 times from current time to next period. Let’s use the well established logistic equation to model the population:
A(n+1) = A(n) + r (1-A(n)/L) x A(n), where A(n) represents population of the rabbits at time n, L represents the carrying capacity of the area (which in this case, is 1500) and r is the restricted growth rate (I.e. 1.5). These values could be found from consistent analysis of experimental data during fieldwork investigations. If A(n)<1500, then 1 - A(n)/L > 0, so at time n+1 the population A(n+1) will increase from A(n) at time n+1 E.g. if A(n) = 500 < 1500, A(n+1) = 1000 and the population has grown by 500. Using this to calculate A(n+2) we find the value is 1500, but if we calculate A(n+3), A(n+4) etc, we find that the value will remain constant at 1500. If we have a population of exactly 1500 rabbits, then A(n)/L = 1 and therefore A(n+1) - A (n) will be 0, and thus there would be no change in population from one period to the next, and we have reached an equilibrium point. In contrast, if A(n) > 1500, (1- A(n)/L) < 0 so population would decrease.
This is an oversimplified general model for some populations where adults often survive after reproduction, and is obviously not suitable for all populations. For example, when it comes to insect populations, it is often the case that they have non-overlapping generations, meaning all adults lay eggs in a specific time of year and then die, leaving only their offspring. To understand the modelling of insect populations we must first understand the Malthusian equation (in discrete time), which is a simple first order model that states Nn+1=ʎNn , where ʎ is the growth rate of the population and Nn is the population at a discrete time n. If b is the average number of births given by any individual and d is the probability of deaths in the population at a given time n, then we could infer that Nn+1 = Nn + Nnb - Nnd = Nn (1+b-d). In the case with non-overlapping generations, if Nn is the population size at given time n, and R0 is the number of offspring per adult, then, ʎ = 1 + R0 - 1, and Nn+1=Nn x R0.
However, you may have realized that this is not a realistic model, as it is unlikely that all offspring will survive to give births to the next generation due to factors such as larvae predation and both intra and inter-specific competitions for limited resources. S(N), the fraction of offspring that survives, is therefore important, and the equation changes to Nn+1 = Nn x S(N) x R0, which for simplicity is generally rewritten as Nn+1 = F(N) x Nn = f(N) (1). It may be interesting to note that F(N) represents the per capita production of offspring that survive to the next generation of a population size N.
If we hypothesize that insect populations are solely affected by intra-specific competitions only, then some further refinement could be done on the model. If there were no competition, then S(N)=1 as all offspring survive. In the second case, if the resources present are not sufficient to support the whole population, then some individuals obtain the units of resources they would need to survive and reproduce, and others die without producing any offspring. Thus in this case S(N) = 1 for Nn < Nc, (where Nc is the critical value above which it would no longer be the case that all individuals survive) and for Nn > Nc, S(N) = Nc/Nn - this type of competition is usually known as contest competitions. The other type of a commonly-known competition is called scramble competition, where every individual is given an equal share of the resources and if Nn < Nc, the critical value, S(N)=1 as all individuals obtain sufficient resources. If Nn > Nc, S(N) = 0 as the amount of resources gained by each individual is not enough to help them survive til the next generation. It is nevertheless very rarely the case that a population, especially a large one, would crash to 0, therefore the behaviour of S(N), and consequently F(N), is always treated as asymptotic.
Idealized contest competition exhibits exact compensation, where the increase in mortality compensates exactly for any increase in population between time n and time n+1. This occurs if F(N)-->c/N as N -->infinity, (where c is a constant) and consequently f(N) -->c, according to (1). This means that over time, the population size would stop fluctuating and remain at value c (as seen in figure 3). Other behaviour could be modelled by F(N)-->c/(N)^b, which infers f(N)-->(N x c)/(N)^b, and if 0 < b < 1, then under-compensation is observed where f(N) will increase over time as number of mortality is always lower than the increase in population. The population at a given time n will be smaller than population at time n+1 after reaching a critical value (as seen in figure 1). On the other hand, if b>1, then the opposite shall be observed where over-compensation takes place and the population at the time n is going to be larger than population at the time n+1 after population has reached the critical value (as seen in figure 2). If F(N) decreases with N, then a compensatory behaviour is said to be observed. As stated above, contest competition happens when b=1, and scramble competition when b>1 so both of these types of competition are compensatory. The non-linear Hassell equation gives a more realistic representation of the population, and could be used to summarise all kinds of compensatory behaviour: Nn+1 = (R0Nn) / (1+aNn)^b where a and R0 are positive constants and b is larger than or equal to 0. When b = 0, there is no competition; when b = 1, there is contest competition and when b>1, there is scramble competition.
These are just a few small glimpses into how populations are monitored. Obviously, there are still limitations to these models, as they do not take into account other factors such as inter-specific competitions, presence of predators etc., however, scientists do hope that the models correspond roughly with real-life data. If that were the case, the models could then become very useful tools when monitoring and predicting the scale of species population trends in the future.