Special Session 36: Complexity in dynamical systems and applications in biology

Impact of Intraspecific Competition of Predator on Coexistence of a Predator-prey Model with Additive Predation on Prey

Dingyong Bai Guangzhou University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we consider a predator-prey model incorporating intraspecific competition in the predator and additive predation on the prey. The additive predation can elicit strong/weak Allee effects in prey, leading to complex dynamics in the model. Our goal is to examine how predator competition affects the species` coexistence. Analyzing the model with the predator competition coefficient as the bifurcation parameter, we reveal the existence and stability of equilibria, and various bifurcation phenomena such as Hopf, saddle-node, and Bogdanov-Takens bifurcations, as well as heteroclinic and homoclinic bifurcations. These bifurcations identify critical thresholds in the competition coefficient, indicating behavioral transitions. Our findings indicate that the strong Allee effect heightens the extinction risk for both species, but predator intraspecific competition reduces this risk and promotes coexistence. Especially when the ratio of the predator`s mortality to conversion rate is low, there are competition thresholds that trigger a range of initial values conducive to coexistence, which widens as competition intensifies. Additionally, our analysis shows that reducing the predation pressure on the prey from other potential predators can foster coexistence by inducing a weak Allee effect or eliminating the Allee effect altogether.

Investigating Multi-Disease Models with Coinfection Coupled with Networks

Christine M Craib University of California, Los Angeles (UCLA) USA Co-Author(s):    Mason A. Porter, Maximillian Eisenberg

Abstract: When considering the effects of related diseases in a population, it is useful to create models that incorporate the dynamics of both diseases. However, it is difficult to derive algebraic expressions for the coexistence of multiple diseases in ODE models. We present an ODE model of 2 non-lethal diseases, each without conferred immunity, with coinfection and universal recovery. Our ODE model assumes homogeneous mixing and instantaneous contacts. We calculate the basic reproductive numbers of each disease and of the system as a whole. We explore all possible equilibria, and we determine necessary and sufficient existence criteria and the linear-stability conditions of those that exist under the conditions of our model. We perform both local and global sensitivity analyses of our model. We relax the ODE assumptions of homogeneous mixing and instantaneous contacts by coupling the ODE system with a contact network. The simplest network maintains the homogeneous-mixing assumption, but it involves prolonged lengths of contact. We then increase network complexity, consider bipartite networks and heterogeneous mixing. We perform local and global sensitivity analyses on all ODE and network parameters and discuss how increasing the complexity of the model affects the projected prevalences of the diseases.

Longtime behavior for solutions to a temporally discrete diffusion equation with a free boundary

Zhiming Guo Guangzhou University Peoples Rep of China Co-Author(s):    Yijie Li, Zhiming Guo, Jian Liu

Abstract: In this talk, we will investigate the longtime behavior of solutions to a temporally discrete diffusion equation with a fixed boundary and a free boundary respectively in one space dimension. Such equation can be equivalent to an integrodifference equation, another important time discrete equation that provides powerful tools for the study of dispersal phenomena. We first discuss the global dynamics of the equation in a fixed bounded domain. With a Stefan type free boundary, we then give a new well-posedness proof and the regular spreading-vanishing dichotomy for the corresponding problem. Moreover, a modified comparison principle for the time discrete free boundary problem is proved in an effort to provide the sufficient conditions for dichotomy. It is the first attempt to study the temporally discrete diffusive phenomenon with a free boundary.

Periodic solutions for second-order difference equations with continuous time

Genghong Lin Guangzhou University Peoples Rep of China Co-Author(s):

Abstract: Due to the essential difficulty of establishing an appropriate variational framework on a suitable working space, how to apply the critical point theory for showing the existence and multiplicity of periodic solutions of continuous-time difference equations remains a completely open problem. New ideas are introduced to overcome such a difficulty. This enables us to employ the critical point theory to construct uncountably many periodic solutions for a class of superlinear continuous-time difference equations without assuming symmetry properties on the nonlinear terms. The obtained solutions are piecewise differentiable in some cases, distinguishing continuous-time difference equations from ordinary differential equations qualitatively. This is a joint work with Zhan Zhou, Zupei Shen, and Jianshe Yu.

Destabilization, stabilization, and multiple attractors in saturated mixotrophic environments

Torsten A Lindstroem Linnaeus University Sweden Co-Author(s):    T. Lindstr\{o}m, Y. Cheng, and S. Chakraborty

Abstract: The ability of mixotrophs to combine phototrophy and phagotrophy is now well recognized and found to have important implications for ecosystem dynamics. In this paper, we examine the dynamical consequences of the invasion of mixotrophs in a system that is a limiting case of the chemostat. The model is a hybrid of a competition model describing the competition between autotroph and mixotroph populations for a limiting resource, and a predator-prey type model describing the interaction between autotroph and herbivore populations. Our results show that mixotrophs are able to invade in both autotrophic environments and environments described by interactions between autotrophs and herbivores. The interaction between autotrophs and herbivores might be in equilibrium or cycle. We find that invading mixotrophs have the ability to both stabilize and destabilize autotroph-herbivore dynamics depending on the competitive ability of mixotrophs. The invasion of mixotrophs can also result in multiple attractors.

Threshold dynamics of a Wolbachia-driven mosquito suppression model on two patches

Xiaoke Ma Harbin Institute of Technology Peoples Rep of China Co-Author(s):    Ying Su

Abstract: Releasing {\it Wolbachia}-infected mosquitoes to control wild mosquitoes is a promising avenue. Many studies have been devoted to using mathematical tools to find the optimal control strategy. However, the impact of diffusion of uninfected/infected mosquitoes is less understood. To describe the discretization of release sites, a two-patch mosquito suppression model with time delay and impulsive release is investigated in this paper. Particularly, we assume that the waiting period between two consecutive releases is equal to the sexual lifespan of infected males. We confirm the well-posedness and monotonicity of the solution and explore the existence and stability of equilibria. For some parameter regimes, we give sufficient conditions for a bistable structure. Based on this, we establish the existence of the separatrix with some sharp estimates when choosing constant functions as initial values. More interestingly, the monotonicity of the separatrix in the release number is proved, implying the existence of an optimal release strategy. We also found that the higher the cytoplasmic incompatibility intensity, the more likely wild mosquitoes are suppressed, and releasing infected males at as many spots as possible is more effective.

Light--scissor--paper: light--mediated intransitivity leads to phytoplankton coexistence

Francesco Paparella New York University Abu Dhabi United Arab Emirates Co-Author(s):    Francesco Paparella (NYUAD); Alessandro Portaluri (UNITO); Xinyue Li (NYUAD)

Abstract: The paradox of plankton (as stated by Hutchinson in 1961) is usually solved by admitting that coexistence equilibria are indeed unstable, but that instability, rather than leading to the extinction of all but one species competing for a single limiting resource, leads to limit cycles (or chaotic attractors) where several species coexist while their populations fluctuate in time. The cyclical nature of multispecies coexistence (already mathematically proven in 1975 by Leonard and May) has later been rationalized as intransitive competition, that is, the inability of any individual competitor to overcompete all of the others, just as in the game rock-scissor-paper. In this work we show that periodic oscillations of a parameter may induce an intransitive competition dynamics, and thus lead to coexistence in situations where only one species would survive if all parameters were constant in time. Specifically, we consider a plankton model proposed by Huisman and Weissing, and extend it to allow for light dependent growth rates, using the Eilers and Peeters growth-irradiance relationship, with realistic parameters fitted from laboratory experiments. We study the case of a single limiting resource, and we deduce necessary and sufficient conditions for the coexistence of two species. We then extend the discussion to the case of an arbitrary number of species.

Periodic solutions for differential equations with distributed delay

Huafeng Xiao Guangzhou University Peoples Rep of China Co-Author(s):    Zhiming Guo, Wieslaw Krawcewicz, Jianshe Yu et. al.

Abstract: In 1934, Volterra poineered the use of an integro-differential equation to model the laboratory populations of some species of small organisms with short generation time. Since then, numerous differential equations incorporating distributed delay have been constructed. Utilizing tools such as fixed point theory, the existence of periodic solutions for these rypes of equations has been established. Recently, Nakata investigated a differential eqution with distributed delay, establishing the existence of at least one periodic solution. However, subsequent numerical simulations have intriguingly revealed the presence of the existence of two periodic solutions. Recognizing the significance of this discovery, our report delves into the matter further, exploring the multiplicity of periodic solutions in differential equations with distributed delay. We achieve this by making use of the critical point theory, equivariant degree theory and Kaplan-Yorke`s method. This is jointed work with Prof. Zhiming Guo, Wieslaw Krawcewicz, Jianshe Yu, et. al.

Geometric theory of distribution shapes for autoregulatory gene circuits

Sheng Ying Guangzhou university Peoples Rep of China Co-Author(s):    Genghong Lin, Feng Jiao, Chen Jia

Abstract: In this study, we provide a complete mathematical characterization of the phase diagram of distribution shapes in an extension of the two-state telegraph model of stochastic gene expression in the presence of positive or negative autoregulation. Using the techniques of second-order difference equations and nonlinear discrete dynamical systems, we prove that the feedback loop can only produce three shapes of steady-state protein distributions (decaying, bell-shaped, and bimodal), corresponding to three distinct parameter regions in the phase diagram. The boundaries of the three regions are characterized by two continuous curves, which can be constructed geometrically by the contour lines of a series of ratio operators. Based on the geometric structure of the phase diagram, we then provide some simple and verifiable sufficient and/or necessary conditions for the existence of the bimodal parameter region, as well as the conditions for the steady-state distribution to be decaying, bell-shaped, or bimodal. Finally, we also investigate how the phase diagram is affected by the strength of positive or negative feedback.

wStri spread dynamics in Nilaparvata lugens via discrete mathematical models

Bo Zheng Guangzhou University Peoples Rep of China Co-Author(s):    Huichao Yang, Saber Elaydi, Jianshe Yu

Abstract: {\it Wolbachia}, an intracellular bacterium, is well-known for inducing cytoplasmic incompatibility, which has become a promising and environmentally sustainable strategy for controlling pest populations. The strain {\it w}Stri, specifically identified in {\it Nilaparvata lugens} (brown planthopper), has shown potential for such biocontrol applications. In this study, we develop a comprehensive discrete mathematical model to analyze the dynamics of {\it w}Stri spread in a mixed population of {\it w}Stri-infected, {\it w}Lug-infected, and uninfected {\it Nilaparvata lugens} under both constant and periodically varying environmental conditions. Under a constant environment, the model identifies the critical threshold necessary for the successful establishment of {\it w}Stri within the population. Our analysis reveals that the model exhibits a strong Allee effect, where a population must exceed a certain critical density, the Allee threshold, for the {\it w}Stri strain to persist and spread. Below this threshold, the {\it w}Stri strain is likely to be eliminated, failing in pest control efforts. When the environment varies periodically, the model transforms into a non-autonomous periodic discrete model, introducing additional complexity. In this scenario, we derive sufficient conditions that ensure the composition of finitely many Allee maps continues to function as an Allee map. Furthermore, we prove that a unique periodic orbit exists within such a periodic environment. This orbit is characterized as unstable and acts as a threshold, determining whether {\it w}Stri will establish itself in the population or die out over time. The findings from this model provide critical insights into the conditions under which {\it w}Stri can be effectively used to control {\it Nilaparvata lugens}, particularly in environments that are not constant but fluctuate periodically. These insights have significant implications for the practical deployment of {\it Wolbachia}-based biocontrol methods in pest management strategies.

Positive solutions for discrete boundary value problems involving the mean curvature operator

Zhan Zhou Guangzhou University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will introduce some results on the positive solutions for some nonlinear discrete Dirichlet boundary value problems involving the mean curvature operator by using critical point theory. Sufficient conditions on the existence of infinitely many positive solutions are given. We show that, the suitable oscillating behavior of the nonlinear term near at the origin and at infinity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions. The existence of at least two positive solutions is also established when the nonlinear term is not oscillatory both at the origin and at infinity. Examples are given to illustrate our main results at last.