Mathematical Models of Pattern Formation in Planktonic Predation-Diffusion Systems: A Review

Robust ecological pattern formation induced by demographic noise

“…[3] have been identified as satisfying, qualitatively at least, the key requirements for diffusion driven pattern formation. Observed patterns of plankton populations have also been proposed to arise from Turing instabilities, at least over short length scales [4,5,6,7].The common feature of these systems is positive feedback coupled to slow diffusion (usually associated with a species labeled an "activator" that activates both itself and another species called the "inhibitor"), and negative feedback coupled to faster diffusion associated with the inhibitor. This combination of diffusion and feedback promotes the formation of patterns, because local patches are promoted through positive feedback, but are only able to spread a limited distance before the fast diffusion and associated negative feedback of the inhibitor prevents further spread.…”

“…One particular class of ecological pattern forming systems, predator-prey (or organism-natural enemy) systems has been extensively analyzed theoretically (see for example, [4,5,8,9,10]) and is beginning to allow qualitative comparison to field data along with more system specific theory [2,11]. A difficulty in directly comparing the results of this large body of theory to field observations is that in many cases, models only exhibit Turing instabilities if the predator diffusivity is much larger than the prey diffusivity or the parameters are fine tuned [4,8,9,11].…”

Fluctuation-driven Turing patterns

“…The final sections of the paper will focus on possible experimental tests and extensions of the theory developed in the body of the paper. We focus on a model of planktonic predator-prey interactions throughout the paper for simplicity and also because predator-prey systems have been extensively analyzed theoretically [11,12,18,20,33] and there is beginning to be an experimental literature [16,19]. However, we emphasize that the goal of this paper is insight into the general interactions of intrinsic fluctuations with the Turing mechanism for pattern formation and that the results should be valid for most models of Turing instabilities.…”

“…Turing's argument, which will be described in detail below, showed how diffusion, which is typically thought of as a randomizing influence, can give rise to spatial pattern formation when there are two or more classes of degrees of freedom (species) with "activator" and "inhibitor" dynamics. This mechanism has been proposed as an explanation for an enormous variety of systems including short (< 10m) length scale patchiness in planktonic ecosystems [11][12][13][14], patterning in plant-resource systems [15], patchiness in insect abundance [16], stripe and spot patterns on the coats of animals [6], patterns in mussel beds [9] and even the geometric visual hallucinations experienced by shamans and users of hallucinogenic drugs [4,17].However, in spite of the seeming success of the Turing mechanism in explaining patterns across many disciplines, the partial differential equations representing the dynamics of systems with Turing patterns typically require unphysical fine tuning of parameters or separation of scales in the diffusivities of the different species in order to predict pattern formation [5,8,11,[18][19][20][21]. The requirement that the system either have fine tuning of kinetic parameters or a separation of scales in diffusivi- * Present Address: Department of Physics and Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge MA, 02139 ties in order to predict patterns, is unphysical for many applications.…”

“…Turing's argument, which will be described in detail below, showed how diffusion, which is typically thought of as a randomizing influence, can give rise to spatial pattern formation when there are two or more classes of degrees of freedom (species) with "activator" and "inhibitor" dynamics. This mechanism has been proposed as an explanation for an enormous variety of systems including short (< 10m) length scale patchiness in planktonic ecosystems [11][12][13][14], patterning in plant-resource systems [15], patchiness in insect abundance [16], stripe and spot patterns on the coats of animals [6], patterns in mussel beds [9] and even the geometric visual hallucinations experienced by shamans and users of hallucinogenic drugs [4,17].…”