Browsing by Subject "pair approximation"
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(2022)Coral reefs form important marine ecosystems and simultaneously are at risk of deterioration due to rapidly changing environments as a consequence of human actions. Understanding their dynamics is thus important in order to be able to protect them from being destroyed. In this thesis we construct a lattice model for two life-history strategies of corals, brooders and spawners. These two strategies differ mainly in their modes of sexual reproduction, but also differences in growth and death rates as well as competitive ability are considered. We use pair approximation to help analyse the model while keeping its spatial structure. Numerical analysis is used to find the equilibria of the system as well as their stabilities, first for a single strategy and then for the two-strategy system. We find that the two strategies are able to coexist if the spawners have a higher growth rate and higher death rate and are competitively superior to brooders. This requires some reproduction over distance and a trade-off between growth and death rates. Thus we find that brooders are focusing a bigger part of their energy on long-distance reproduction, while spawners are dominating over short distances and having a higher turnover. We also find that both mutual invasibility and coexistence in the broader sense are only possible for low rates of sexual reproduction for both strategies. For higher rates of sexual reproduction we find that whichever strategy invades the lattice first will stay and the other cannot invade. Lastly we look at the effect of a change in environmental conditions, namely the acidification and temperature increase of oceans, on the two strategies and find that it affects the two strategies differently. The spawners are quickly driven to extinction by the change in environmental conditions, while brooders initially benefit from the changing conditions and only start to suffer themselves after the spawners have gone extinct.
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(2024)Altruism refers to behavior by an individual that increases the fitness of another individual while decreasing their own, and despite seemingly going against traditional theories of evolution, it's actually quite common in the animal kingdom. Understanding why and how altruistic behaviors happen has long been a central focus in evolutionary ecology, and this thesis aims to contribute to this area of study. This thesis focuses on infinite lattice models. Lattice models are a type of spatially explicit models, which means that they describe the dynamics of a population in both time and space. In particular, we consider a modification of the simplest type of lattice models (called the contact process), which considers only birth and death events. The objective is to study altruistic behaviours to help neighbours within populations residing on a lattice. To achieve this, we assume that, apart from giving birth and dying, individuals transition to a permanently non-reproductive state at a certain rate. We use ordinary differential equations to describe the dynamics of this population and to develop our model. The population we initially have in the lattice (the resident population) reaches a positive equilibrium, which we calculate numerically using Matlab. Trough linear stability analysis, we can show that this equilibrium is asymptotically stable, which means that with time, the resident population will stabilize at this equilibrium. Once the resident reaches this equilibrium, we introduce a mutant population in the lattice with the same characteristics as the resident, except that it has a different post-reproductive death rate. Linear stability analysis of the extinct equilibrium of the mutant shows that mutants with a higher post-reproductive death rate than the residents gain a competitive advantage. This is because by dying faster, post-reproductive mutants make more space for other mutants to reproduce. That result changes if we make the assumption that post-reproductive individuals help their neighbours produce more offspring. In this case, we find that depending on the amount of reproductive help given by the post-reproductive individuals, a higher post-reproductive death rate no longer is evolutionary advantageous. In fact, we are able to determine that, in general, helping neighbours reproduce is a better strategy than sacrificing oneself to make room for reproductive neighbours. Lastly, we examine this reproductive help as a function of the post-reproductive mortality rate. With this, our goal is to find an evolutionary stable strategy (ESS) for the resident population, that is, a strategy that cannot be displaced by any alternative strategies.
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