Reaction Populations of Preys (P) and Predators (Q) Lotka-Volterra Model and = constant coefficients
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
Understanding the equation for the preys The growth rate per capta is interspecies competition Allee effect logistic growth decrease because of predators P
Understanding the equation for the predators The growth rate per capta is mortality rate + interspecies competition logistic growth increase because of preys Q
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
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
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
The new equations describing the dynamics of Preys (P) and Predators (Q) become local interactions diffusion diffusion coefficients Partial Differential Equations
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!
2 - Look for solutions that are constant in time:
Two solutions: 1 – extinction Q = P = 0 2 – coexistence Q 0 = 10 P 0 = 5. Are they stable?
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
Predator-Prey systems on a Network: Two main difficulties: 1 – describe diffusion in the network 2 – do the stability analysis
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
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
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
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
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.
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.