Chaotic provinces in the kingdom of the Red Queen

The interplay between parasites and their hosts is found in all kinds of species and plays an important role in understanding the principles of evolution and coevolution. Usually, the different genotypes of hosts and parasites oscillate in their abundances. The well-established theory of oscillatory Red Queen dynamics proposes an ongoing change in frequencies of the different types within each species. So far, it is unclear in which way Red Queen dynamics persists with more than two types of hosts and parasites. In our analysis, an arbitrary number of types within two species are examined in a deterministic framework with constant or changing population size. This general framework allows for analytical solutions for internal fixed points and their stability. For more than two species, apparently chaotic dynamics has been reported. Here we show that even for two species, once more than two types are considered per species, irregular dynamics in their frequencies can be observed in the long run. The nature of the dynamics depends strongly on the initial configuration of the system; the usual regular Red Queen oscillations are only observed in some parts of the parameter region

Host-parasite coevolution with mulitiple types

The coevolution of hosts and parasites has been analysed most prominently with two types using deterministic, population-based models. These models usually generate oscillatory (Red Queen) dynamics. So far it was unclear in which way Red Queen dynamics persists with more than two types of hosts and parasites. In stochastic models changing population size reduces the probability of Red Queen dynamics in a model with two types. It was also argued that with more types in a stochastic model Red Queen dynamics can be observed in a limited parameter space which decreases as the number of host and parasite types increases. In this thesis an arbitrary number of types is examined using deterministic methods. A xed point and stability analysis is conducted and constants of motions are formulated. We show that Red Queen dynamics can still exist. However, Hamiltonian chaos is possible in large areas of the parameter space.Die Koevolution von Wirten und Parasiten wird meist mit zwei Arten mittels deterministischen, populationsbasierten Modellen analysiert. Diese Modelle erzeugen in der Regel oszillierende (Red Queen) Dynamiken. Bisher war unklar, in welcher Weise Red Queen Dynamiken bei mehr als zwei Arten von Wirten und Parasiten bestehen bleiben. In stochastischen Modellen reduziert eine veranderliche Populationsgroe die Wahrscheinlichkeit von Red Queen Dynamik in einem Modell mit zwei Arten. Es wurde auerdem diskutiert, dass mit mehr Arten in einem stochastischen Modell Red Queen Dynamik in einem begrenzten Parameteraum besteht. Dieser Parameterraum reduziert sich, je hoher die Anzahl der Wirt und Parasit Typen. In dieser Arbeit wird, unter Verwendung von deterministischen Methoden, eine beliebige Anzahl an Arten von Wirten und Parasiten untersucht. Eine Fixpunkt- und Stabilitatsanalyse wird durchgefuhrt und Bewegungskonstanten werden formuliert. Wir zeigen, dass Red Queen Dynamik noch existiert. Jedoch entsteht in groen Bereichen des Parameterraums Hamilton'sches Chaos.Contents 1 Introduction 7 2 Mathematical methods 9 2.1 Interaction models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Matching allele model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Cross-infection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 General infection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Replicator dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Single population dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.2 Two population dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Lotka-Volterra dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Fixed points and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.6 Constant of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.7 Analysis of chaotic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.8 Stochastic simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Analytical results 17 3.1 Fixed points and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.1 Replicator dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.2 Lotka-Volterra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Constant of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1 Replicator dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.2 Lotka-Volterra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Numerical analysis 31 4.1 Three types and chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Stochastic simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5 Discussion 37 6 Appendix 41 6.1 Jacobian entries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.2 Constant of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Constraint satisfaction mechanisms for marginal stability and criticality in large ecosystems

We discuss a resource-competition model, which takes the MacArthur's model as a platform, to unveil interesting connections with glassy features and jamming in high dimension. This model presents two qualitatively different phases: a "shielded" phase, where a collective and self-sustained behavior emerges, and a "vulnerable" phase, where a small perturbation can destabilize the system and contribute to population extinction. We first present our perspective based on a strong similarity with continuous constraint satisfaction problems in their convex regime. Then, we discuss the stability in terms of the computation of the leading eigenvalue of the Hessian matrix of the free energy in the replica space. This computation allows us to efficiently distinguish between the two aforementioned phases and to relate high-dimensional critical ecosystems to glassy phenomena in the low-temperature regime.Comment: Updated version with references added. 6 pages, 2 figure

The Predator's Numerical and Functional Responses Derived from First Principles : Population Dynamical and Evolutionary Consequences

This article-based dissertation uses mathematical models to study predator-prey interactions and their population dynamical and evolutionary consequences. The focus is on the predator's numerical and functional response which I derive from first principles, i.e., from the interactions between prey and predator individuals. The aim is to connect population-level phenomena and the long-term evolution of the prey or the predator to processes on the level of the individuals. The dissertation consists of a general introductory part and three research articles with general results as well as applications to specific models. The first two articles have already been published in the Journal of Mathematical Biology and in the Journal of Theoretical Biology, respectively. The third article is under review for publication. In the first research article, I introduce a formal method for the derivation of a predator's functional and numerical response from the interactions between the individual prey and predators. Such derivation permits an explicit interpretation of the parameters and structure of the functional and numerical responses in terms of individual behaviour. The general method is illustrated with several concrete examples. Some examples give novel derivations of already well-known functional responses. Other examples give derivations for responses that have not been used before and lead to a rich population dynamical behaviour including Allee effects as well as simultaneous existence of multiple positive population-dynamical attractors. In the second research article, I model a stand-off between a predator and a prey individual when the prey is hiding and the predator is waiting for the prey to come out from its refuge, or when the two are locked in a situation of mutual threat of injury or even death. The stand-off is resolved when the predator gives up or when the prey tries to escape. Using the methods of the first article, this individual-level model leads to the well-known Rosenzweig-MacArthur model but now with parameters that directly connect to the behaviour of the individuals, in particular the giving-up rates of the prey and the predator. I use the model to study the coevolution of the giving-up rates using the mathematical theory of adaptive dynamics. New and different evolutionary results emerge in comparison with the asymmetric war of attrition in evolutionary game theory which is the more traditional way of modelling a stand-off. In the third research article, I study the evolution of density dependent handling times (i.e., the processing time of captured prey) and the related functional and numerical responses. It is a well-established theoretical result that coexistence of two predator species feeding on one and the same prey is possible, but only if the system exhibits non-equilibrium dynamics. Coexistence is possible because the two predator species occupy different temporal niches: the one with the longer handling time has the advantage when the prey is rare so that holding on to the same catch is the better option, while the species with the shorter handling time has the advantage when the prey is common and easy to catch. Using the adaptive dynamics approach, I show that a predator species with a non-constant handling time that decreases with the prey density is selectively superior regardless of whether the prey is rare or common. The reason is that such generalist predator can occupy both temporal niches all by itself. By means of these examples, the dissertation demonstrates the strengths of deriving population models from first principles as it enables us to connect population-level phenomena and long-term evolution to the behaviour of the individuals that make up the population.Tämä artikkelipohjainen väitöskirja tutkii matemaattisten mallien avulla petoeläimen ja saaliin välistä vuorovaikutusta ja niiden populaation dynaamisia ja evolutiivisia seurauksia. Työn painopiste on petoeläimen numeerisessa ja toiminnallisessa vasteessa, jonka johdan ensimmäisistä periaatteista eli saaliin ja petoeläinten välisistä vuorovaikutuksista. Tavoitteena on yhdistää populaatiotason ilmiöt ja saaliin tai petoeläimen pitkäaikainen kehitys yksilötason prosesseihin. Väitöskirja koostuu yleisestä johdanto-osasta ja kolmesta tutkimusartikkelista, joissa on yleisiä tuloksia sekä sovelluksia tiettyihin malleihin. Kaksi ensimmäistä artikkelia on jo julkaistu Journal of Mathematical Biology ja Journal of Theoretical Biology -lehdissä. Kolmatta artikkelia tarkastellaan julkaisua varten. Ensimmäisessä tutkimusartikkelissa esittelen muodollisen menetelmän saalistajan toiminnallisen ja numeerisen vasteen johtamiseksi yksittäisen saaliin ja petoeläinten välisistä vuorovaikutuksista. Tällainen johtaminen mahdollistaa funktionaalinen vaste ja numeeristen vasteiden parametrien ja rakenteen eksplisiittisen tulkinnan yksilön käyttäytymisen kannalta. Yleistä menetelmää havainnollistetaan useilla konkreettisilla esimerkeillä. Jotkut esimerkit antavat uusia johdannaisia jo hyvin tunnetuista toiminnallisista vasteista. Muut esimerkit antavat johdannaisia vastauksista, joita ei ole käytetty aiemmin ja jotka johtavat runsaaseen populaation dynaamiseen käyttäytymiseen, mukaan lukien Allee-vaikutukset sekä useiden useampien dynaamisten attraktorien samanaikainen olemassaolo samanaikainen olemassaolo. Toisessa tutkimusartikkelissa mallinnan eroa saalistajan ja saalisyksilön välillä, kun saalis on piilossa ja saalistaja odottaa saaliin tulevan ulos turvapaikastaan tai kun nämä kaksi ovat lukittuina tilanteeseen, jossa on molemminpuolinen loukkaantumisen tai jopa kuoleman uhka. Vastakkainasettelu ratkeaa, kun saalistaja luovuttaa tai kun saalis yrittää paeta. Ensimmäisen artikkelin menetelmiä käyttäen tämä yksilötason malli johtaa hyvin tunnettuun Rosenzweig-MacArthurin -malliin, mutta nyt parametreillä, jotka liittyvät suoraan yksilöiden käyttäytymiseen, erityisesti saaliin ja saalistajan luopumisasteeseen. Mallin avulla tutkin pedon ja saaliin välisen luovuttamisen yhteisevoluutiota adaptiivisen dynamiikan viitekehyksen avulla. Minun mallini tuottamat uudet evolutiiviset tulokset poikkeavat merkittävästi evoluutiopeliteorian tutkimuksessa tyypillisesti käytetyn näännytyssodan mallin tuloksista. Kolmannessa tutkimusartikkelissa tutkin saalispopulaatioiden tiheyksistä riippuvien käsittelyaikojen (eli pyydystetyn saaliin käsittelyajan) evoluutiota ja niihin liittyviä funktionaalisia ja numeerisia vasteita. Vakiintunut teoreettinen tulos kertoo kahden samaa saalista syövän saalislajin rinnakkaiselon olevan mahdollista vain, jos populaatiodynaaminen tila on epätasapainossa. Rinnakkaiselo on mahdollista, koska kaudella saalislajilla voi olla erilainen ekologinen lokero: pidemmän käsittelyajan omaavalla on etulyöntiasema, kun saalis on harvinainen, jolloin pyydystetyn saaliin syömisessä pitäytyminen on parempi vaihtoehto, kun taas lyhyemmän käsittelyajan omaavalla lajilla on etulyöntiasema, kun saalis on yleinen ja helppo pyydystää. Adaptiivisen dynamiikan viitekehystä käyttämällä osoitan, että saalistajalaji, jonka käsittelyaika ei ole vakio ja joka pienenee saalistiheyden mukana, on valikoivasti parempi riippumatta siitä, onko saalis harvinainen vai yleinen. Syynä on, että tällainen yleispetoeläin voi varata molemmat ekologiset lokerot itselleen. Väitöskirja osoittaa näillä esimerkeillä populaatiomallien johtamisen vahvuuksia ensimmäisistä periaatteista, koska se mahdollistaa populaatiotason ilmiöiden ja pitkän aikavälin evoluution yhdistämisen populaation muodostavien yksilöiden käyttäytymiseen

The evolution and dynamics of interacting populations.

SIGLEAvailable from British Library Document Supply Centre- DSC:DX173189 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Complex systems in Ecology: a guided tour with large Lotka-Volterra models and random matrices

Ecosystems represent archetypal complex dynamical systems, often modelled by coupled differential equations of the form $$\frac{d x_i}{d t} = x_i \varphi_i(x_1,\cdots, x_N)\ ,$$ where $N$ represents the number of species and $x_i$, the abundance of species $i$. Among these families of coupled diffential equations, Lotka-Volterra (LV) equations $$\frac{d x_i}{d t} = x_i ( r_i - x_i +(\Gamma \mathbf{x})_i)\ ,$$ play a privileged role, as the LV model represents an acceptable trade-off between complexity and tractability. Here, $r_i$ represents the intrinsic growth of species $i$ and $\Gamma$ stands for the interaction matrix: $\Gamma_{ij}$ represents the effect of species $j$ over species $i$. For large $N$, estimating matrix $\Gamma$ is often an overwhelming task and an alternative is to draw $\Gamma$ at random, parametrizing its statistical distribution by a limited number of model features. Dealing with large random matrices, we naturally rely on Random Matrix Theory (RMT). The aim of this review article is to present an overview of the work at the junction of theoretical ecology and large random matrix theory. It is intended to an interdisciplinary audience spanning theoretical ecology, complex systems, statistical physics and mathematical biology

Evolutionary Dynamics of Predator-Prey Systems: An Ecological Perspective

Evolution takes place in an evolutionary setting that typically involves interactions with other organisms. To describe such evolution, a structure is needed which incorporates the simultaneous evolution of interacting species. Here a formal framework for this purpose is suggested, extending from the microscopic interactions between individuals- the immediate cause of natural selection, through the mesoscopic population dynamics responsible for driving the replacement of one mutant phenotype by another, to the macroscopic process of phenotypic evolution arising from many such substitutions. The process of coevolution that results from this is illustrated in the predator-prey systems. With no more than qualitative information about the evolutionary dynamics, some basic properties of predator-prey coevolution become evident. More detailed understanding requires specification of an evolutionary dynamic; two models for this purpose are outlined, one from our own research on a stochastic process of mutation and selection and the other from quantitative genetics. Much of the interest in coevolution has been to characterize the properties of fixed points at which there is no further phenotypic evolution. Stability analysis of the fixed points of evolutionary dynamical systems is reviewed and leads to conclusions about the asymptotic states of evolution rather than different from those of game-theoretic methods. These differences become especially important when evolution involves more than one species

Order out of Randomness : Self-Organization Processes in Astrophysics

Self-organization is a property of dissipative nonlinear processes that are governed by an internal driver and a positive feedback mechanism, which creates regular geometric and/or temporal patterns and decreases the entropy, in contrast to random processes. Here we investigate for the first time a comprehensive number of 16 self-organization processes that operate in planetary physics, solar physics, stellar physics, galactic physics, and cosmology. Self-organizing systems create spontaneous {\sl order out of chaos}, during the evolution from an initially disordered system to an ordered stationary system, via quasi-periodic limit-cycle dynamics, harmonic mechanical resonances, or gyromagnetic resonances. The internal driver can be gravity, rotation, thermal pressure, or acceleration of nonthermal particles, while the positive feedback mechanism is often an instability, such as the magneto-rotational instability, the Rayleigh-B\'enard convection instability, turbulence, vortex attraction, magnetic reconnection, plasma condensation, or loss-cone instability. Physical models of astrophysical self-organization processes involve hydrodynamic, MHD, and N-body formulations of Lotka-Volterra equation systems.Comment: 61 pages, 38 Figure