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Instabilidades de Turing e Competição Aparente em Ambientes Fragmentados Marcus A.M. de Aguiar Lucas Fernandes IFGW - Unicamp

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Vegetation in arid ecosystems Padrões de densidade em regiões homogêneas

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Wolfram’s Cellular Automata

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After 250 iterations

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Complex Patterns can emerge out of simple interactions between neighboring cells on a homogeneous environment Cellular automata provide examples where direct interactions occur

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Another Mechanism: Reaction-Diffusion Systems and Turing Patterns 1912-1954

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Summary 1.Turing Patterns in Homogeneous Environments 2.Turing Patterns in Networks of Fragments 3.Turing Patterns and Apparent Competition in Networks

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1. Turing Patterns in Homogeneous Environments

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Reaction Populations of Preys (P) and Predators (Q) Lotka-Volterra Model and = constant coefficients

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The Mimura-Murray model On a diffusive prey-predator model which exhibits patchiness M. Mimura and J.D. Murray J. Ther. Biol. (1978) 75, 249-262

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Understanding the equation for the preys The growth rate per capta is interspecies competition Allee effect logistic growth decrease because of predators P

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Understanding the equation for the predators The growth rate per capta is mortality rate + interspecies competition logistic growth increase because of preys Q

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Add a new ingredient: space P(x,t) = population of preys at spatial position x at time t Q(x,t) = population of predators at spatial position x at time t Add a new interaction : migration The populations at x change by sending a fraction of its individuals to neighboring sites and receiving a fraction of individuals from the same sites. Diffusion

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The math of diffusion kk-1k+1 P(k) – P(k-1) = 2 P(k-1) = 5 P(k) = 7 P(k+1) = 8 P(k+1) – P(k) = 1 A gradient in P(k) is not enough: diffusion requires a gradient of the gradient back

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A uniform gradient is not enough kk-1k+1 P(k) – P(k-1) = 2 P(k-1) = 5 P(k) = 7 P(k+1) = 9 P(k+1) – P(k) = 2

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The new equations describing the dynamics of Preys (P) and Predators (Q) become local interactions diffusion diffusion coefficients Partial Differential Equations

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Can we understand these equations? Are there simple solutions? 1 - Look for solutions that are uniform in space, i.e., situations where the populations are the same at all points in space. In this case there is no diffusion!

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2 - Look for solutions that are constant in time:

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Simplified equations can be solved:

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Two solutions: 1 – extinction Q = P = 0 2 – coexistence Q 0 = 10 P 0 = 5. Are they stable?

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Patterns of preys and predators emerging on a homogeneous environment. Preys distributed on patches. Predators everywhere, but with larger populations where the preys live. J. Theor. Biol. 1978 u=P=prey v=Q=pred

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2. Turing Patterns in Networks of Fragments

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Predator-Prey systems on a Network: Two main difficulties: 1 – describe diffusion in the network 2 – do the stability analysis

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Describe the network: 1 - labels the nodes from 1 to N in order of decreasing number of connections. 2 – Define the N x N adjancency matrix A ij = 1 if nodes i and j are connected A ij = 0 if nodes i and j are not connected 3 – k i = number of connections of node i

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Examples: 5 1 2 3 4 1 2 3 4 5

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Model diffusion: Diffusion matrix Diffusion of u at node i where u i is the population at node i

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Example 1: 5 1 2 3 4 For node 2: diffusion

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Example 2: For node 1: 1 2 3 4 5

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New Notation: u = prey = P v = predator = Q For zero diffusion we are back to the same equations, for which there is a homogeneous solution: each community has the same number of preys and predators. We find u i = u 0 = 5 and v i = v 0 = 10 for all nodes uv predator prey each node is a fragment, a local community

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Movies turn on diffusion:

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Hysteresis

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3. Turing Patterns and Apparent Competition in Networks u x v y predatorspreys

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Results Dynamics for first prey, u Dynamics for second prey, x Dynamics for u-x Parameters: =0.12 =20.0 =0.05 f=0.5 BA network with N=1000

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Effect of Coupling

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Three nested predator-prey pairs in each node uv xy predatorspreys wz Typical patterns: sites with v-u and z-w and low values of y-x sites with y-x and low values of v-u and z-w few sites with all species in equal proportions

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Four predator-prey pairs in each node uv xy predatorspreys wz Typical patterns: sites with v-u and z-w and low values of y-x and r-s sites with y-x and r-s and low values of v-u and z-w few sites with all species in equal proportions rs

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Conclusions on a homogeneous environment, density patterns can be generated dynamically, independent of intrinsic differences. on a fragmented environment with identical patches, abundance distributions can be different: there will be two types of patches: with high abundance and with low abundance.

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if more pairs of antagonistic species interact in each patch, strong effects of apparent competition can also be dinamically generated. There will be four types of patches: - high v and u with low y and x - high y and x with low v and u - low v, u, y and x. - high v, u, y and x.

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Two-species competition The Lotka-Volterra Model Working with differential equations to predict population dynamics.

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