Luiz Alberto Díaz Rodrigues

Pattern formation, long-term transients, and the Turing-Hopf bifurcation in a space- and time-discrete predator-prey system.

Understanding of population dynamics in a fragmented habitat is an issue of considerable importance. A natural modelling framework for these systems is spatially discrete. In this paper, we consider a predator–prey system that is discrete both in space and time, and is described by a Coupled Map Lattice (CML). The prey growth is assumed to be affected by a weak Allee effect and the predator dynamics includes intra-specific competition. We first reveal the bifurcation structure of the corresponding non-spatial system. We then obtain the conditions of diffusive instability on the lattice. In order to reveal the properties of the emerging patterns, we perform extensive numerical simulations. We pay a special attention to the system properties in a vicinity of the Turing–Hopf bifurcation, which is widely regarded as a mechanism of pattern formation and spatiotemporal chaos in space-continuous systems. Counter-intuitively, we obtain that the spatial patterns arising in the CML are more typically stationary, even when the local dynamics is oscillatory. We also obtain that, for some parameter values, the system’s dynamics is dominated by long-term transients, so that the asymptotical stationary pattern arises as a sudden transition between two different patterns. Finally, we argue that our findings may have important ecological implications.

Pattern formation in a space- and time-discrete predator–prey system with a strong Allee effect

The spatiotemporal dynamics of a space- and time-discrete predator–prey system is considered theoretically using both analytical methods and computer simulations. The prey is assumed to be affected by the strong Allee effect. We reveal a rich variety of pattern formation scenarios. In particular, we show that, in a predator–prey system with the strong Allee effect for prey, the role of space is crucial for species survival. Pattern formation is observed both inside and outside of the Turing domain. For parameters when the local kinetics is oscillatory, the system typically evolves to spatiotemporal chaos. We also consider the effect of different initial conditions and show that the system exhibits a spatiotemporal multistability. In a certain parameter range, the system dynamics is not self-organized but remembers the details of the initial conditions, which evokes the concept of long-living ecological transients. Finally, we show that our findings have important implications for the understanding of population dynamics on a fragmented habitat.

Patchy Invasion of Stage-Structured Alien Species with Short-Distance and Long-Distance Dispersal

Understanding of spatiotemporal patterns arising in invasive species spread is necessary for successful management and control of harmful species, and mathematical modeling is widely recognized as a powerful research tool to achieve this goal. The conventional view of the typical invasion pattern as a continuous population traveling front has been recently challenged by both empirical and theoretical results revealing more complicated, alternative scenarios. In particular, the so-called patchy invasion has been a focus of considerable interest; however, its theoretical study was restricted to the case where the invasive species spreads by predominantly short-distance dispersal. Meanwhile, there is considerable evidence that the long-distance dispersal is not an exotic phenomenon but a strategy that is used by many species. In this paper, we consider how the patchy invasion can be modified by the effect of the long-distance dispersal and the effect of the fat tails of the dispersal kernels.

A mathematical model for growth in weight of silver catfish (Rhamdia quelen) (Heptapteridae, Siluriformes, Teleostei)

The use of a mathematical model applied to biological science helps to predict the specific data. Based on biological data (weight and age) of silver catfish, Rhamdia quelen, a mathematical model was elaborated based on a nonlinear difference equation to demonstrate the relationship between age and growth in weight. Silver catfish growth was described following the Beverton-Holt model Pt+1 = (r Pt) / (1+ a Pt ), where r > 0 is the maximum growth rate and a > 0 is a constant of growth inhibition. The solution of this equation is Pt= 1 /{[1/P0 - a / (r-1)] 1/rt + a/ (r-1)}, were P0 is the initial weight of the fish. Through this model it was observed that the female reaches the theoretical maximum weight approximately at the age of 18 years and the male at the age of 12 years in a natural environment.

Coupled Map Lattice Model for Insects and Spreadable Substances

Understanding the spreading dynamics of insects and particles naturally or artificially associated with them, such as seeds, pollen, repellents or insecticides, is of paramount importance for pest management and conservation programs. Insects and chemical or natural products exhibit dispersal patterns that depend on the environment where they are and their respective sizes. In this chapter, we present a Coupled Map Lattice formalism to investigate the theoretical dynamics of the spread of insects and chemical substances sprayed over them. The models consider a habitat with abundant resources and therefore insects moving only in response to chemical concentrations. Diffusion and wind are the mechanisms used to spread chemical substances. Continuous and discrete models are used to describe the system on a macroscopic scale. The results are discussed taking into account rules for movement, escape behaviour and integrated pest management strategies.