Darwin illustrated his theory about emergence and evolution of biological species with adiagram. It shows how species exist, evolve, appear and disappear. The goal of this workis to give a mathematical interpretation of this diagram and to show how it can bereproduced in mathematical models. It appears that conventional models in populationdynamics are not sufficient, and we introduce a number of new models which take intoaccount local, nonlocal and global consumption of resources, and models with space andtime dependent coefficients.

In this paper we explore the eco-evolutionary dynamics of a predator-prey model, wherethe prey population is structured according to a certain life history trait. The traitdistribution within the prey population is the result of interplay between geneticinheritance and mutation, as well as selectivity in the consumption of prey by thepredator. The evolutionary processes are considered to take place on the same time scaleas ecological dynamics, i.e. we consider the evolution to be rapid. Previously publishedresults show that population structuring and rapid evolution in such predator-prey systemcan stabilize an otherwise globally unstable dynamics even with an unlimited carryingcapacity of prey. However, those findings were only based on direct numerical simulationof equations and obtained for particular parameterizations of model functions, whichobviously calls into question the correctness and generality of the previous results. Themain objective of the current study is to treat the model analytically and considervarious parameterizations of predator selectivity and inheritance kernel. We investigatethe existence of a coexistence stationary state in the model and carry out stabilityanalysis of this state. We derive expressions for the Hopf bifurcation curve which can beused for constructing bifurcation diagrams in the parameter space without the need for adirect numerical simulation of the underlying integro-differential equations. Weanalytically show the possibility of stabilization of a globally unstable predator-preysystem with prey structuring. We prove that the coexistence stationary state is stablewhen the saturation in the predation term is low. Finally, for a class of kernelsdescribing genetic inheritance and mutation we show that stability of the predator-preyinteraction will require a selectivity of predation according to the life trait.

A reaction–diffusion replicator equation is studied. A novel method to apply theprinciple of global regulation is used to write down a model with explicit spatialstructure. Properties of stationary solutions together with their stability are analyzedanalytically, and relationships between stability of the rest points of thenon-distributed replicator equation and the distributed system are shown. In particular,we present the conditions on the diffusion coefficients under which the non-distributedreplicator equation can be used to describe the number and stability of the stationarysolutions to the distributed system. A numerical example is given, which shows that thesuggested modeling framework promotes the system’s persistence, i.e., a scenario ispossible when in the spatially explicit system all the interacting species survive whereassome of them go extinct in the non-distributed one.

An overview of a general approach for mathematical modeling of evolving heterogeneouspopulations using a wide class of selection systems and replicator equations (RE) ispresented. The method allows visualizing evolutionary trajectories of evolvingheterogeneous populations over time, while still enabling use of analytical tools ofbifurcation theory. The developed theory involves introducing escort systems of auxiliary“keystone" variables, which reduce complex multi-dimensional inhomogeneous models tolow dimensional systems of ODEs that in many cases can be investigated analytically. Inaddition to a comprehensive theoretical framework, a set of examples of the method’sapplicability to questions ranging from preventing the tragedy of the commons to cancertherapy is presented.

Nothing is more fundamental to life than the ability to reproduce and duplicate theinformation cells store in their genomes. The mechanism of duplication of DNA has beenconserved from prokaryotes to eukaryotes. The aim of the study was to quantify whichevolutionary forces could produce the pattern of genome replication architecture observedin present-day organisms. This was achieved using an evolutionary simulation, combiningrandom genome sequence shuffling, mutation, selection and the mathematical modeling of DNAreplication. We have found parameter values which explained evolutionary pressures of DNAreplication in E.coli, P.calidifontis and S.cerevisae. Surprisingly, the results of the evolutionary simulation suggeststhat for a fixed cost per replication origin it is more advantageous for genomes to reducethe number of replication origins under increasing uncertainty in origin activationtiming.

In mammalian cells, the p53 pathway regulates the response to a variety of stresses,including oncogene activation, heat and cold shock, and DNA damage. Here we explore amathematical model of this pathway, composed of a system of partial differentialequations. In our model, the p53 pathway is activated by a DNA-compromising event of shortduration. As is typical for mathematical models of the p53 pathway, our model contains anegative feedback loop representing interactions between the p53 and Mdm2 proteins. Anovel feature of our model is that we combine a spatio-temporal approach with theappearance and repair of DNA damage. We investigate the behaviour of our model throughnumerical simulations. By ignoring the possibility of DNA repair, we first explore thescenario in which the cell has a very inefficient DNA repair mechanism. We find thatspatio-temporal oscillations in p53 and Mdm2 may occur, consistent with experimental data.We then allow p53 to be directly involved in repairing DNA damage, since experimentalevidence suggests this can happen. We find that oscillations in p53 and Mdm2 can stilloccur, but their amplitude damps down quickly as the DNA damage is repaired. Finally, wefind that a minor change to the location of the DNA damage can notably change the spatialdistribution of p53 within the nucleus. We discuss the biological implications of ourresults.

Amazon molly (Poecilia formosa) females reproduce asexually, but theyneed sperm to initiate the process. Such gynogenetic reproduction can be called spermparasitism since the DNA in the sperm is not used. Since all offspring of asexuallyreproducing females are females, they can locally outcompete sexually reproducing ones,but their persistence is threatened by the lack of males. Therefore, the existence ofAmazon mollies is puzzling. A metapopulation structure has been suggested to enable thecoexistence of gynogenetic and sexual species. Previously only Levins-type metapopulationmodels have been used to investigate this question, but they are not defined on theindividual level. Therefore we investigate the evolution of sperm parasitism in astructured metapopulation model, which incorporates both realistic local populationdynamics and individual-level dispersal. If the reproduction strategy is freely evolvingin a large well-mixed population or in the structured metapopulation model, strongdiscrimination of asexually reproducing females by males results in evolution to fullsexuality, whereas mild discrimination leads to too small probability of sexualreproduction, so that the lack of males causes the extinction of the evolving population,resulting in evolutionary suicide. This classification remains the same also when bothsexual reproduction and dispersal are freely evolving. Sexual and asexual behaviour can beobserved at the same time in this model in the presence of a trade-off between thereproduction and dispersal traits. However, we do not observe disruptive selectionresulting in the evolutionarily stable coexistence of fully sexual and fully asexualfemales. Instead, the presence of sexual and asexual behaviour is due to females with amixed reproduction trait.

We introduce a game theoretical model of stealing interactions. We model the situation asan extensive form game when one individual may attempt to steal a valuable item fromanother who may in turn defend it. The population is not homogeneous, but rather eachindividual has a different Resource Holding Potential (RHP). We assume that RHP not onlyinfluences the outcome of the potential aggressive contest (the individual with the largerRHP is more likely to win), but that it also influences how an individual values aparticular resource. We investigate several valuation scenarios and study the prevalenceof aggressive behaviour. We conclude that the relationship between RHP and resource valueis crucial, where some cases lead to fights predominantly between pairs of strongindividuals, and some between pairs of weak individuals. Other cases lead to no fightswith one individual conceding, and the order of strategy selection is crucial, where theindividual which picks its strategy first often has an advantage.

The majority of species are under predatory risk in their natural habitat and targeted bypredators as part of the food web. During the evolution of ecosystems, manifold mechanismshave emerged to avoid predation. So called secondary defences, which are used after apredator has initiated prey-catching behaviour, commonly involve the expression of toxinsor deterrent substances which are not observable by the predator. Hence, the possession ofsuch secondary defence in many prey species comes with a specific signal of that defence(aposematism). This paper builds on the ideas of existing models of such signallingbehaviour, using a model of co-evolution and generalisation of aversive information andintroduces a new methodology of numerical analysis for finite populations. This newmethodology significantly improves the accessibility of previous models.

In finite populations, investigating the co-evolution of defence and signalling requiresan understanding of natural selection as well as an assessment of the effects of drift asan additional force acting on stability. The new methodology is able to reproduce thepredicted solutions of preceding models and finds additional solutions involving negativecorrelation between signal strength and the extent of secondary defence. In addition,genetic drift extends the range of stable aposematic solutions through the introduction ofa new pseudo-stability and gives new insights into the diversification of aposematicdisplays.