Mathematical biology is commonly referred to as a division of biology that relies on mathematical models and theoretical analysis. These models areused to explain biological factorsusing mathematical representation, which researchers hope to display in order to showaccurate results based ontheir data.Within mathematical biology numerous peer-reviewed journal articles have been published, I have chosen to research the mathematics behindthe population dynamics of animals.In this articlethe researchers talk about marine mega vertebrates or more specifically about the population of lemon sharks, which due to their frequent migrationmakes them problematicto study.
Within this study mathematical models were developed to helpaccess factors that influence the population dynamics of marine mega vertebrates. Within this article the factors they have decided are important when looking for population trends are “...age-structure, density-dependent feedbacks on reproduction, and demographic stochasticity” (White, Nagy, Gruber). The reasonthey developed these mathematical models are becausemarine mega vertebrates often play a key rolein our ecosystems makingtheir populationsvery importantto manage.In order to model thefactors that influencelemon shark population a Markov-chain process structured around age was used. Markov-chain relates to a sequence of random variables, which allows the researchers to consider the intricate life cycle of lemon sharks which entails many variables.Throughout the study the lifespan used is 0 to 26, this is due to the maximum age of the sharks is thought to be around 25. In order to use their model, they first had to find the estimates of fecundity and mortality among the population of lemon sharks. To calculate fecundity two equations are used, one accounts for the number of females that give birth in a year, while the other describes probability density for the total population.
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Forthe first equation to work we assume that there is anequal sex ratio among females and males that are born, While, the second equation is based offPoisson distribution (typically used for birth processes) and is going off two assumptions. One assumption beinga fixed mean among the Poisson distribution and the second is each probability density is already set for females to produceiamount of offspring. When using the equation, R represents arandom variable taking on a set of values.Then density dependent mortality is calculated which is represented by a Michaelis-Menten function. Which overall is the representation of the number of sharks that die between the first and second census of the population.
Overall these equationsand the article in generalrequiresextensive amounts of knowledge in probabilityto understand and thoroughly review. This relates back to why they used a Markov-chain processto compensate for the vast number of variables used when calculating probabilities. Lastly, they implement model which is what required estimates of fecundity and mortality for it to be related to the factors that affect thepopulation of lemon sharks. This output allows them to compare values from the field work conducted in Bimini to get potential values, or what they call the Monte Carlo technique.When looking at their findings they compare their models results to those find in the field work. Which suggest that around 77 juvenile sharks (age 0-2) live in the lagoon at a time, which their model correctly predicts. However, their model’s variance was off, while the model’s distribution of variance was 176 it only account for 35% of the random sample.
To summarize the stochastic model when used looked at the interactions of the factors that influence population, in which all these factors accounted for 33%-49% of variance in the lemon shark’s population size.Population fluctuations can be caused by many factors, which the results of this model being 33% to 49% variance in the shark’s population size only accounts for their tested factors. Interpreting this information, we may be able to infer the rest of the variance may be natural or environmental factors not tested within the model. When looking back at the purpose of this model we can see that it should only be used to broadly speculate factors that influence population dynamics due to its somewhat impreciseness. Or more so it doesn’t account for all factors that influence population dynamics. Without a large amount of education in probability and statistics this article is very hard to comprehend and review. However, the only thing I can recommend without extensive amounts of knowledge in the subject is to try other types of models to fit their data rather than just one type to compare accurateness.
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