Using non-smooth models to determine thresholds for microbial pest management

A joint-threshold Filippov model describing the effect of intermittent androgen-deprivation therapy in controlling prostate cancer

Intermittent androgen-deprivation therapy (IADT) can be beneficial to delay the occurrence of treatment resistance and cancer relapse compared to the standard continuous therapy. To study the effect of IADT in controlling prostate cancer, we developed a Filippov prostate cancer model with a joint threshold function: therapy is implemented once the total population of androgen-dependent cells (AC-Ds) and androgen-independent cells (AC-Is) is greater than the threshold value ET, and it is suspended once the population is less than ET. As the parameters vary, our model undergoes a series of sliding bifurcations, including boundary node, focus, saddle, saddle-node and tangency bifurcations. We also obtained the coexistence of one, two or three real equilibria and the bistability of two equilibria. Our results demonstrate that the population of AC-Is can be contained at a predetermined level if the initial population of AC-Is is less than this level, and we choose a suitable threshold value.

A modeling framework for adaptive collective defense: crisis response in social-insect colonies

Living systems, from cells to superorganismic insect colonies, have an organizational boundary between inside and outside and allocate resources to defend it. Whereas the micro-scale dynamics of cell walls can be difficult to study, the adaptive allocation of workers to defense in social-insect colonies is more conspicuous. This is particularly the case for Tetragonisca angustula stingless bees, which combine different defensive mechanisms found across other colonial animals: (1) morphological specialization (distinct soldiers (majors) are produced over weeks); (2) age-based polyethism (young majors transition to guarding tasks over days); and (3) task switching (small workers (minors) replace soldiers within minutes under crisis). To better understand how these timescales of reproduction, development, and behavior integrate to balance defensive demands with other colony needs, we developed a demographic Filippov ODE system to study the effect of these processes on task allocation and colony size. Our results show that colony size peaks at low proportions of majors, but colonies die if minors are too plastic or defensive demands are too high or if there is a high proportion of quickly developing majors. For fast maturation, increasing major production may decrease defenses. This model elucidates the demographic factors constraining collective defense regulation in social insects while also suggesting new explanations for variation in defensive allocation at smaller scales where the mechanisms underlying defensive processes are not easily observable. Moreover, our work helps to establish social insects as model organisms for understanding other systems where the transaction costs for component turnover are nontrivial, as in manufacturing systems and just-in-time supply chains.

The state-dependent impulsive control for a general predator-prey model

In this paper, a general predator-prey model with state-dependent impulse is considered. Based on the geometric analysis and Poincaré map or successor function, we construct three typical types of Bendixson domains to obtain some sufficient conditions for the existence of order-1 periodic solutions. At the same time, the existing domains are discussed with respect to the system parameters. Moreover, the Analogue of Poincaré Criterion is used to obtain the asymptotic stability of the periodic solutions. Finally, to illustrate the results, an example is presented and some numerical simulations are carried out.

Bifurcations in discontinuous mathematical models with control strategy for a species

In this paper a preliminary mathematical model is proposed, by means of a system of ordinary differential equations, for the growth of a species. In this case, the species does not interact with another species and is divided into two stages, those that have or have not reached reproductive maturity, with natural and control mortality for both stages. When performing a qualitative analysis to determine conditions in the parameters that allow the extinction or preservation of the species, a modification is made to the model when only control is assumed for each of the stages if the number of species in that stage is above a critical value. These studies are carried out by bifurcation analysis with respect to two parameters: control for each stage and their critical values. It is concluded that for certain conditions in their parameters, the dynamics in each of the controlled stages converge to their critical values.

Dynamic behaviors of the FitzHugh-Nagumo neuron model with state-dependent impulsive effects

In present work, in order to reproduce spiking and bursting behavior of real neurons, a new hybrid biological neuron model is established and analyzed by combining the FitzHugh-Nagumo (FHN) neuron model, the threshold for spike initiation and the state-dependent impulsive effects (impulse resetting process). Firstly, we construct Poincaré mappings under different conditions by means of geometric analysis, and then obtain some sufficient criteria for the existence and stability of order-1 or order-2 periodic solution to the impulsive neuron model by finding the fixed point of Poincaré mapping and some geometric analysis techniques. Numerical simulations are given to illustrate and verify our theoretical results. The bifurcation diagrams are presented to describe the phenomena of period-doubling route to chaos, which implies that the dynamic behavior of the neuron model become more complex due to impulsive effects. Furthermore, the correctness and effectiveness of the proposed FitzHugh-Nagumo neuron model with state-dependent impulsive effects are verified by circuit simulation. Finally, the conclusions of this paper are analyzed and summarized, and the effects of random factors on the electrophysiological activities of neuron are discussed by numerical simulation.