Abstracts
Spatio-Temporal Patterns in Simple Models of Marine Systems
U. Feudel
University of Oldenburg
Spatio-temporal patterns in marine systems are a result of the interaction of population dynamics with physical transport processes. These physical transport processes can be either diffusion processes in marine sediments or advection of biological species in the water column. We study in a simplified model the dynamics of one population of bacteria and its nutrient in sediments, taking into account that the considered bacteria possess an active as well as an inactive state, where activation is processed by signal molecules. Furthermore the nutrients are transported actively by bioirrigation and passively by diffusion. It is shown that under certain conditions Turing patterns can occur which yield heterogeneous spatial patterns of species. The influence of bioirrigation on Turing patterns leads to the emergence of ''hot spots``, i.e. localized regions of enhanced bacterial activity. In the water column advection is the dominant physical process. We study the influence of mesoscale hydrodynamic structures on biological growth processes in the wake of an island. Using a stream function approach for the velocity field we show how the upwelling of nutrients away from the island affects the evolution of plankton close to it. In particular we show that mesoscale vortices act as incubators for plankton growth leading to localized plankton blooms within vortices.
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Finite-dimensional asymptotics and emergence of optimality in general infinite systems with inheritance
A. N. Gorban
University of Leicester
In the 1970s to the 1980s, theoretical work developed a “common” field simultaneously applicable to physics, biology and mathematics. For physics it is (so far) part of the theory of a special kind of approximation, demonstrating, in particular, interesting mechanisms of discreteness in the course of the evolution of distributions with initially smooth densities. However, what for physics is merely a convenient approximation is a fundamental law in biology: inheritance. The consequences of inheritance (collected in the selection theory).
Consider a community of animals. Let it be biologically isolated. Mutations can be neglected in the first approximation. In this case, new genes do not emerge.
An example from physics is as follows. Let waves with wave vectors k be excited in some system. Denote K a set of wave vectors k of excited waves. Let the wave interaction does not lead to the generation of waves with new k (not in K). Such an approximation is applicable to a variety of situations, and has been described in detail for wave turbulence.
What is common in these examples is the evolution of a distribution with a support that does not increase over time. What does not increase must, as a rule, decrease, if the decrease is not prohibited. This naive thesis is converted into rigorous theorems for the case under consideration. It is proved that the support decreases in the limit of infinite time if it was sufficiently large initially. (At finite times the distribution supports are conserved and decrease only in the limit.) Conservation of the support usually results in the following effect: dynamics of an initially infinite-dimensional system on for large time intervals can be described by finite-dimensional systems and specific optimality principles.
A non-linear kinetic system with conservation of supports for distributions generically has (under some additional technical conditions) finite-dimensional asymptotics. This conservation of support has a quasi-biological interpretation: inheritance (if a gene is not present initially in an isolated population without mutations, then it cannot appear at a later time).
The finite-dimensional asymptotics demonstrates effects of “natural” selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become increasingly narrow over time and move increasingly slowly. It is possible that these peaks do not tend to fixed positions, and the path covered tends to infinity as with time growth. The drift equations for peak motion are obtained.
Various types of distribution stability are studied: internal stability (stability with respect to perturbations that do not extend the support), external stability or uninvadability (stability with respect to strongly small perturbations that extend the support), and stable realizability (stability with respect to small shifts and extensions of the density peaks). The general condition for stable realizability is a stability condition with respect to the corresponding drift equations (a finite system of ordinary differential equations).
Models of self-synchronization of cell division are studied, as an example. Kinetic systems with conservation of supports are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics.
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A Maximum Entropy Closure for Spatial Moments in Spatial Ecology
N. Hill
University of Glasgow
When using stochastic spatial point processes to model the dispersal and distribution of plants, e.g. trees in rain forests, we derive equations for the spatial moments that describe the average density and clustering of the plants. The equations for any one moment involve higher order moments so, to solve them, we need a scheme for closing the system. In this work, I shall describe an approach that maximises the entropy of higher order moments to achieve a closure scheme.
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Consumer-Resource Spatial Dynamics in Realistic Landscapes
A. H. Hirzel (1), R. M. Nisbet (2), W. W. Murdoch (2)
(1) University of Lausanne, (2) University of California at Santa Barbara
This work explores the effect of spatial processes in a heterogeneous environment on the dynamics of a specialist consumer-resource interaction.
The environment consists of a lattice of favourable (habitat) and hostile (matrix) hexagonal cells, whose spatial distribution is measured by habitat proportion and spatial autocorrelation (inverse of fragmentation). At each time step, a fixed fraction of both populations disperses to the adjacent cells where it reproduces following the Nicholson-Bailey model.
Aspects of the dynamics analysed include extinction, stability, cycle period and amplitude, and the spatial patterns emerging from the dynamics.
Depending primarily on the fraction of the resource population that disperses in each generation and on the landscape geometry, five classes of spatio-temporal dynamics can be objectively distinguished, which we call syndromes: Spatial chaos, Spirals, Metapopulation, Mainland-island and Spiral fragments. The two firsts are commonly found in theoretical studies of homogeneous landscapes. The three others are direct consequences of the heterogeneity and have strong similarities to dynamic patterns observed in real systems (e.g. extinction-recolonisation, source-sink, outbreaks, spreading waves).
The importance of these results is threefold: First, our model merges into a same theoretical framework dynamics commonly observed in the field that are usually modelled independently. Second, these dynamics and patterns are explained by dispersal rate and common landscape statistics, thus linking in a practical way population ecology to landscape ecology. Third, we show that the landscape geometry has a qualitative effect on the length of the cycles and, in particular, we demonstrate how very long periods can be produced by spatial processes.
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The Evolution of Altruism Through Beard Chromodynamics
V. A. A. Jansen and M. van Baalen
University of London, Royal Holloway
The evolution of altruism and cooperation poses a long-standing evolutionary paradox. One of the solutions that has been proposed is recognition: if individuals could direct their altruistic behaviour to other altruists, thus avoiding being exploited by selfish, non-altruistic individuals altruistic and cooperative behaviour could, in theory, evolve. The simplest recognition system is a conspicuous, heritable tag, for example a green beard (the example is due to Richard Dawkins). It has, however, been argued that the 'green beard effect' is practically implausible, despite the fact that some examples of such genes are known. We have modelled the green beard effect and found that if the recognition tags are diverse, altruism is easily facilitated. Beard colour diversity allows altruism to evolve and be maintained. We have called the resulting evolutionary dynamics "beard chromodynamics" because many beard colours co-occur in dynamic mosaic in the population.
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Lineage Integrity in Neutral Evolution and Temporal Intermittency in Co-Evolution
H. J. Jensen
Imperial College London
Although mutation prone reproduction of individuals in a neutral type space is very similar to random walks, at least one striking difference exists. Namely a collection of random walkers disperse, i.e. densities peaks disappears. A density peak of reproducing individuals in type space on the other hand is able to remain a peak, while the peak itself performs a random walk. We discuss a theory of the diffusing peak. When interaction is introduced between the co-evolving individuals, the dynamics at the level of macroevolution becomes intermittent and quasi-stable epochs are punctuated by periods of rapid collective adaptive search. We present the phenomenology of a model of co-evolution, the Tangled Nature model, and discuss the intermittent dynamics in terms of record statistics.
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Spatiotemporal Patterns in Deterministic and Stochastic Predation-Diffusion Systems With Infected Prey
H. Malchow
University of Osnabrück
A model of prey-predator dynamics is considered for the case of infection of the prey population. Pathogens (V) are taken into account either as constant control parameter or by an explicit equation. The prey population (P) is split into a susceptible (S) and an infected (I) part. Both parts grow logistically, limited by a common carrying capacity. The predator (Z) is feeding on susceptibles and infected, following a Holling-type II and III functional response, respectively. The local analysis of the (V-)S-I-Z differential equations yields a number of stationary and/or oscillatory regimes. In space, one finds waves of recovery or invading infection like diffusive fronts, target patterns and spiral waves. Correspondingly interesting is the behaviour under multiplicative noise, modelled by stochastic partial differential equations. The external noise can enhance the survival of susceptibles and infected, respectively, that would go extinct in a deterministic environment. In the parameter range of excitability, noise can induce local and global prey-predator oscillations as well as local coherence resonance and global synchronization.
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Understanding the Spatiotemporal Dynamics of Rodent Populations: Multiannual Cycles and Periodic Travelling Waves
M. Smith (1), J. Sherratt (1), X. Lambin (2)
(1) Heriot-Watt University, (2) University of Aberdeen
Rodent populations that show regular cycles in abundance with a period of several years are one of the most fascinating and widely studied phenomena in ecology. I will talk about our own investigations into the cyclic Field Vole populations in Kielder Forest, Northern England. I will aim to cover a variety of past and present empirical and theoretical studies into why these populations cycle and why the spatiotemporal dynamics in the past showed periodic travelling waves in abundace.
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On Some Models in Population Dynamics With Nonlocal Consumption of Resources
V. Volpert
University of Lyon
We will discuss integro-differential equations and systems of equations arising in population dynamics with nonlocal consumption of resources. The nonlocal terms are related to the intra-specific competition. These models can be applied to study the emergence and the evolution of biological species and to some other biological questions. They provide an interesting nonlinear dynamics with pattern formation and propagation of complex waves.
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