Impacts Of Intraspecific Diffusion Variation On Fish Spatial Pattern Formations

Patterns are some regular but inhomogeneous structures defined on a specific time or spatial scale,which are ubiquitous in nature.The animate world provides a great number of examples such as the heterogeneous species distribution.Researchers found that the discrepancy of diffusion rates between preys and predators is an important factor driving the emergence of spatial patterns.However,the impact of intraspecific diffusion variation caused by individual ontogenetic development on pattern formations is still barely known.In the present work,we construct a consumer-resource model,and propose four diffusion modes for the size-structured juvenile predators,to explore how the intraspecific diffusion variation affects the spatio-temporal dynamics.The work is structured as follows.Chapter 1:We mainly describe the classic mechanisms triggering pattern emergence,including the Turing instability,Hopf bifurcation,Turing-Hopf bifurcation,and pattern study under the same framework of predator-prey population models.We further state the main contents of this paper.Chapter 2:We construct a consumer-resource temporal model where the juvenile predator is size-structured to explore the system dynamics,including the existence,uniqueness and stability of equilibrium.We first derive the steady state,provide the numerical schemes,and analyse the time-series solution of species biomass.Then,we apply the bifurcation theory to the temporal model so that we can figure out how the resource carrying capability K and adult competitive ability q(two important model parameters)influence the community dynamics.Finally,we explore the stable region spanned by these two parameters mentioned above.Chapter 3:We extend the size-structured temporal model in Chapter 2 to a size-structured spatio-temporal counterpart by allowing difusion for both species.We propose four size-dependent diffusion modes(hump-model,uphill-mode,Logistic-mode,and downhill-mode)for the juvenile consumers.We future introduce the coefficient of variation(CV)as a quantitative measure of spatial aggregation of emergent spatial patterns.At last,we explore the sensitivity of this index with respect to the initial value.Chapter 4:This chapter explores how symmetric diffusion(i.e.juveniles have an average diffusion rate the same as the adults),asymmetric diffusion between juveniles and adults under different diffusion modes affect the spatio-temporal patterns.Results show that in the case of symmetric diffusion,the intraspecific diffusion variance among juveniles can promote the emergence of patterns.Moreover,spatial patterns can occur when the juveniles and adults possess asymmetric diffusion rates even if it is impossible in the case of symmetric diffusion.Considering that the individual ontogeny is a widespread scenario in nature,the intraspecific diffusion variance may be an important and common mechanism triggering the pattern formations.Finally,we make a summary and put forward some problems needed to be studied in the future.