Ezio Venturino

Effect of delay in a Lotka–Volterra type predator–prey model with a transmissible disease in the predator species

We consider a system of delay differential equations modeling the predator-prey ecoepidemic dynamics with a transmissible disease in the predator population. The time lag in the delay terms represents the predator gestation period. We analyze essential mathematical features of the proposed model such as local and global stability and in addition study the bifurcations arising in some selected situations. Threshold values for a few parameters determining the feasibility and stability conditions of some equilibria are discovered and similarly a threshold is identified for the disease to die out. The parameter thresholds under which the system admits a Hopf bifurcation are investigated both in the presence of zero and non-zero time lag. Numerical simulations support our theoretical analysis.

Patterns of Patchy Spread in Deterministic and Stochastic Models of Biological Invasion and Biological Control

A few spatiotemporal models of population dynamics are considered in relation to biological invasion and biological control. The patterns of spread in one and two spatial dimensions are studied by means of extensive numerical simulations. We show that, in the case that population multiplication is damped by the strong Allee effect (when the population growth rate becomes negative for small population density), in a certain parameter range the spread can take place not via the intuitively expected circular expanding population front but via motion and interaction of separate patches. Alternatively, the patchy spread can take place in a system without Allee effect as a result of strong environmental noise. We then show that the phenomenon of deterministic patchy invasion takes place ‘at the edge of extinction’ so that a small change of controlling parameters either brings the species to extinction or restores the travelling population fronts. Moreover, we show that the regime of patchy invasion in two spatial dimensions actually takes place when the species go extinct in the corresponding 1-D system.

Spiders as biological controllers in the agroecosystem.

In this paper, we propose a general model consisting of insects, pests and spiders interacting in an agroecosystem included in a typical homogeneous rural landscape, characterized by a continuous mosaic of cultivated land and a few small patches of grasslands and small woods bounding the fields. The model is general enough to show all the phenomena observed in the agroecosystem. The role of the spider population as a biological controller in the agroecosystem is particularly emphasized. Human intervention by means of pesticide spraying and its relationship with the biological pest controllers is also accounted for.

A MINIMAL MODEL FOR ECOEPIDEMICS WITH GROUP DEFENSE

The new idea of group defense as recently introduced by the author in the context of two interacting populations is in this paper applied to communities subject also to a disease. The system is formulated with the bare minimum of interactions among all the populations involved in order to highlight the effects of the nonlinearity describing the defense mechanism. A key parameter identified in the purely demographic model, which completely describes its outcomes, is seen here to have an important role also, in that it is dropping below a threshold prevents the disease from invading the environment and causes the healthy prey and predators to coexist via persistent oscillations.

Reliable approximation of separatrix manifolds in competition models with safety niches

In dynamical systems saddle points partition the domain into basins of attractions of the remaining locally stable equilibria. This situation is rather common especially in population dynamics models, like prey–predator or competition systems. Focusing on squirrels population models with niche, in this paper we design algorithms for the detection and the refinement of points lying on the separatrix manifold partitioning the phase space. We consider both the two populations and the three populations cases. To reconstruct the separatrix curve and surface, we apply the Partition of Unity method, which makes use of Wendlands functions as local approximants.

Patchy agglomeration as a transition from monospecies to recurrent plankton blooms

We propose a model for explaining both red tides and recurring phytoplankton blooms. Three assumptions are made, namely the presence of toxin producing phytoplankton, the satiation phenomenon in zooplanktons feeding, modelled by a Holling type II response, and phytoplankton aggregation leading to formation of patches. The dynamics of the plankton population is shown to depend on the fraction of the phytoplankton population that aggregates to form colonies and on the number of the latter.

Simple Metaecoepidemic Models

We consider a simple predator-prey system with two possible habitats and where an epidemic spreads by contact among the prey, but it cannot affect the predators. Only the prey population can freely move from one environment to another. Several models are studied, for different assumptions on the structure of the demographic interactions and on the predators’ feeding. Some counterintuitive results are derived. The role the safety refuge may in some cases entail negative consequences for the whole ecosystem. Also, depending on the system formulation, coexistence of all the populations may not always be supported.

Geometric modeling and motion analysis of the epicardial surface of the heart

Pathological processes cause abnormal regional motions of the heart. Regional wall motion analyses are important to evaluate the success of therapy, especially of cell therapy, since the recovery of the heart in cell therapy proceeds slowly and results in only small changes of ventricular wall motility. The usual ultrasound imaging of heart motion is too inaccurate to be considered as an appropriate method. MRI studies are more accurate, but insufficient to reliably detect small changes in regional ventricular wall motility. We thus aim at a more accurate method of motion analysis. Our approach is based on two imaging modalities, viz. cardiac CT and biplane cineangiography. The epicardial surface represented in the CT data set at the end of the diastole is registered to the three-dimensionally reconstructed epicardial artery tree from the angiograms in end-diastolic position. The motion tracking procedures are carried out by applying thin-plate spline transformations between the epicardial artery trees belonging to consecutive frames of our cineangiographic imagery.

Robust Approximation Algorithms for the Detection of Attraction Basins in Dynamical Systems

A particular solution of a dynamical system is completely determined by its initial condition. When the omega limit set reduces to a point, the solution settles at steady state. The possible steady states of the system are completely determined by its parameters. However, with the same parameter set, it is possible that several steady states can originate from different initial conditions (multi-stability). In that case the outcome depends on the chosen initial condition. Therefore, it is important to assess the domain of attraction for each possible attractor. The algorithms presented here are general and robust enough so as to solve the problem of reconstructing the basin of attraction of each stable equilibrium point. In order to have a graphical representation of the separatrix manifolds, we focus on systems of two and three ordinary differential equations exhibiting bi- or tri-stability. For this purpose we have implemented several Matlab functions for the approximation of the points lying on the curves or on the surfaces determining the basins of attraction and for the reconstruction of such curves and surfaces. We approximate the latter with the implicit partition of unity method using radial basis functions as local approximants. Numerical results, obtained with a Matlab package made available to the scientific community, support our findings.

Global stability and persistence in LG–Holling type II diseased predator ecosystems

A Leslie–Gower–Holling type II model is modified to introduce a contagious disease in the predator population, assuming that disease cannot propagate to the prey. All the system’s equilibria are determined and the behaviour of the system near them is investigated. The main mathematical issues are global stability and bifurcations for some of the equilibria, together with sufficient conditions for persistence of the ecosystem. Counterintuitive results on the role played by intraspecific competition are highlighted.