629 research outputs found
Dynamics of prey–predator model with strong and weak Allee effect in the prey with gestation delay
This study proposes two prey–predator models with strong and weak Allee effects in prey population with Crowley–Martin functional response. Further, gestation delay of the predator population is introduced in both the models. We discussed the boundedness, local stability and Hopf-bifurcation of both nondelayed and delayed systems. The stability and direction of Hopfbifurcation is also analyzed by using Normal form theory and Center manifold theory. It is shown that species in the model with strong Allee effect become extinct beyond a threshold value of Allee parameter at low density of prey population, whereas species never become extinct in weak Allee effect if they are initially present. It is also shown that gestation delay is unable to avoiding the status of extinction. Lastly, numerical simulation is conducted to verify the theoretical findings. 
Qualitative Analysis of a Modified Leslie-Gower Predator-prey Model with Weak Allee Effect II
The article aims to study a modified Leslie-Gower predator-prey model with Allee effect II, affecting the functional response with the assumption that the extent to which the environment provides protection to both predator and prey is the same. The model has been studied analytically as well as numerically, including stability and bifurcation analysis. Compared with the predator-prey model without Allee effect, it is found that the weak Allee effect II can bring rich and complicated dynamics, such as the model undergoes to a series of bifurcations (Homoclinic, Hopf, Saddle-node and Bogdanov-Takens). The existence of Hopf bifurcation has been shown for models with (with- out) Allee effect and the local existence and stability of the limit cycle emerging through Hopf bifurcation has also been studied. The phase portrait diagrams are sketched to validate analytical and numerical findings
Qualitative Analysis of a Modified Leslie-Gower Predator-prey Model with Weak Allee Effect II
The article aims to study a modified Leslie-Gower predator-prey model with Allee effect II, affecting the functional response with the assumption that the extent to which the environment provides protection to both predator and prey is the same. The model has been studied analytically as well as numerically, including stability and bifurcation analysis. Compared with the predator-prey model without Allee effect, it is found that the weak Allee effect II can bring rich and complicated dynamics, such as the model undergoes to a series of bifurcations (Homoclinic, Hopf, Saddle-node and Bogdanov-Takens). The existence of Hopf bifurcation has been shown for models with (without) Allee effect and the local existence and stability of the limit cycle emerging through Hopf bifurcation has also been studied. The phase portrait diagrams are sketched to validate analytical and numerical findings
Turing instability in a diffusive predator-prey model with multiple Allee effect and herd behavior
Diffusion-driven instability and bifurcation analysis are studied in a
predator-prey model with herd behavior and quadratic mortality by incorporating
multiple Allee effect into prey species. The existence and stability of the
equilibria of the system are studied. And bifurcation behaviors of the system
without diffusion are shown. The sufficient and necessary conditions for Turing
instability occurring are obtained. And the stability and the direction of Hopf
and steady state bifurcations are explored by using the normal form method.
Furthermore, some numerical simulations are presented to support our
theoretical analysis. We found that too large diffusion rate of prey prevents
Turing instability from emerging. Finally, we summarize our findings in the
conclusion
Spatiotemporal dynamics of a diffusive predator–prey model with fear effect
This paper concerned with a diffusive predator–prey model with fear effect. First, some basic dynamics of system is analyzed. Then based on stability analysis, we derive some conditions for stability and bifurcation of constant steady state. Furthermore, we derive some results on the existence and nonexistence of nonconstant steady states of this model by considering the effect of diffusion. Finally, we present some numerical simulations to verify our theoretical results. By mathematical and numerical analyses, we find that the fear can prevent the occurrence of limit cycle oscillation and increase the stability of the system, and the diffusion can also induce the chaos in the system
Novel Dynamics in an Additional Food provided Predator-Prey System with mutual interference
The provision of additional food (AF) sources to an introduced predator has
been identified as a mechanism to improve pest control. However, AF models with
prey dependent functional responses can cause unbounded growth of the predator
\cite{S27}. To avoid such dynamics, an AF model with mutual interference effect
has been proposed \cite{S02}. The analysis therein reveals that if the quantity
of additional food , where is the mutual
interference parameter, then pest eradication is possible, and this is
facilitated via a transcritical bifurcation. We revisit this model and show
novel dynamical behaviors. In particular, pest eradication is possible for a
tighter range of AF , and can
also occur via a saddle node bifurcation. We observe bi-stability, as well as
local bifurcations of Hopf type. We also prove a global bifurcation, of
homoclinic type. This bifurcation in turn is shown to create a non-standard
dynamic wherein the pest extinction state becomes an ``almost" global
attractor. To the best of our knowledge, this is the first proof of existence
of such a dynamical structure in AF models. We discuss our analysis in the
context of designing novel bio-control strategies
STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN ECOLOGY AND EPIDEMICS
Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and predictions in various areas of life sciences, such as population dynamics, epidemiology, immunology, physiology, and neural networks. The memory or time-delays, in these models, are related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on. In ordinary differential equations (ODEs), the unknown state and its derivatives are evaluated at the same time instant. In DDEs, however, the evolution of the system at a certain time instant depends on the past history/memory. Introduction of such time-delays in a differential model significantly improves the dynamics of the model and enriches the complexity of the system.
Moreover, natural phenomena counter an environmental noise and usually do not follow deterministic laws strictly but oscillate randomly about some average values, so that the population density never attains a fixed value with the advancement of time. Accordingly, stochastic delay differential equations (SDDEs) models play a prominent role in many application areas including biology, epidemiology and population dynamics, mostly because they can offer a more sophisticated insight through physical phenomena than their deterministic counterparts do. The SDDEs can be regarded as a generalization of stochastic differential equations (SDEs) and DDEs.This dissertation, consists of eight Chapters, is concerned with qualitative and quantitative features of deterministic and stochastic delay differential equations with applications in ecology and epidemics. The local and global stabilities of the steady states and Hopf bifurcations with respect of interesting parameters of such models are investigated. The impact of incorporating time-delays and random noise in such class of differential equations for different types of predator-prey systems and infectious diseases is studied. Numerical simulations, using suitable and reliable numerical schemes, are provided to show the effectiveness of the obtained theoretical results.Chapter 1 provides a brief overview about the topic and shows significance of the study. Chapter 2, is devoted to investigate the qualitative behaviours (through local and global stability of the steady states) of DDEs with predator-prey systems in case of hunting cooperation on predators. Chapter 3 deals with the dynamics of DDEs, of multiple time-delays, of two-prey one-predator system, where the growth of both preys populations subject to Allee effects, with a direct competition between the two-prey species having a common predator. A Lyapunov functional is deducted to investigate the global stability of positive interior equilibrium. Chapter 4, studies the dynamics of stochastic DDEs for predator-prey system with hunting cooperation in predators. Existence and uniqueness of global positive solution and stochastically ultimate boundedness are investigated. Some sufficient conditions for persistence and extinction, using Lyapunov functional, are obtained. Chapter 5 is devoted to investigate Stochastic DDEs of three-species predator prey system with cooperation among prey species. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution to the model are established, by constructing a suitable Lyapunov functional. Chapter 6 deals with stochastic epidemic SIRC model with time-delay for spread of COVID-19 among population. The basic reproduction number ℛs0 for the stochastic model which is smaller than ℛ0 of the corresponding deterministic model is deduced. Sufficient conditions that guarantee the existence of a unique ergodic stationary distribution, using the stochastic Lyapunov functional, and conditions for the extinction of the disease are obtained. In Chapter 7, some numerical schemes for SDDEs are discussed. Convergence and consistency of such schemes are investigated. Chapter 8 summaries the main finding and future directions of research.
The main findings, theoretically and numerically, show that time-delays and random noise have a significant impact in the dynamics of ecological and biological systems. They also have an important role in ecological balance and environmental stability of living organisms. A small scale of white noise can promote the survival of population; While large noises can lead to extinction of the population, this would not happen in the deterministic systems without noises. Also, white noise plays an important part in controlling the spread of the disease; When the white noise is relatively large, the infectious diseases will become extinct; Re-infection and periodic outbreaks can also occur due to the time-delay in the transmission terms
Normal form for singular Bautin bifurcation in a slow-fast system with Holling type III functional response
Over the last few decades, complex oscillations of slow-fast systems have
been a key area of research. In the theory of slow-fast systems, the location
of singular Hopf bifurcation and maximal canard is determined by computing the
first Lyapunov coefficient. In particular, the analysis of canards is based on
the genericity condition that the first Lyapunov coefficient must be non-zero.
This manuscript aims to further extend the results to the case where the first
Lyapunov coefficient vanishes. For that, the analytic expression of the second
Lyapunov coefficient and the investigation of the normal form for codimension-2
singular Bautin bifurcation in a predator-prey system is done by explicitly
identifying the locally invertible parameter-dependent transformations. A
planar slow-fast predator-prey model with Holling type III functional response
is considered here, where the prey population growth is affected by the weak
Allee effect, and the prey reproduces much faster than the predator. Using
geometric singular perturbation theory, normal form theory of slow-fast
systems, and blow-up technique, we provide a detailed mathematical
investigation of the system to show a variety of rich and complex nonlinear
dynamics including but not limited to the existence of canards, relaxation
oscillations, canard phenomena, singular Hopf bifurcation, and singular Bautin
bifurcation. Additionally, numerical simulations are conducted to support the
theoretical findings
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