31 research outputs found
Unveiling the dynamics of canard cycles and global behaviour in a singularly perturbed predator-prey system with Allee effect in predator
In this article, we have considered a planar slow-fast modified Leslie-Gower
predator-prey model with a weak Allee effect in the predator, based on the
natural assumption that the prey reproduces far more quickly than the predator.
We present a thorough mathematical analysis demonstrating the existence of
homoclinic orbits, heteroclinic orbits, singular Hopf bifurcation, canard limit
cycles, relaxation oscillations, the birth of canard explosion by combining the
normal form theory of slow-fast systems, Fenichel's theorem and blow-up
technique near non-hyperbolic point. We have obtained very rich dynamical
phenomena of the model, including the saddle-node, Hopf, transcritical
bifurcation, generalized Hopf, cusp point, homoclinic orbit, heteroclinic
orbit, and Bogdanov-Takens bifurcations. Moreover, we have investigated the
global stability of the unique positive equilibrium, as well as bistability,
which shows that the system can display either 'prey extinction', 'stable
coexistence', or 'oscillating coexistence' depending on the initial population
size and values of the system parameters. The theoretical findings are verified
by numerical simulations.Comment: 26 pages, 13 figure
Evolutionary Suicide of Prey : Matsuda and Abrams' Model Revisited
Under the threat of predation, a species of prey can evolve to its own extinction. Matsuda and Abrams (Theor Popul Biol 45:76-91, 1994a) found the earliest example of evolutionary suicide by demonstrating that the foraging effort of prey can evolve until its population dynamics cross a fold bifurcation, whereupon the prey crashes to extinction. We extend this model in three directions. First, we use critical function analysis to show that extinction cannot happen via increasing foraging effort. Second, we extend the model to non-equilibrium systems and demonstrate evolutionary suicide at a fold bifurcation of limit cycles. Third, we relax a crucial assumption of the original model. To find evolutionary suicide, Matsuda and Abrams assumed a generalist predator, whose population size is fixed independently of the focal prey. We embed the original model into a three-species community of the focal prey, the predator and an alternative prey that can support the predator also alone, and investigate the effect of increasingly strong coupling between the focal prey and the predator's population dynamics. Our three-species model exhibits (1) evolutionary suicide via a subcritical Hopf bifurcation and (2) indirect evolutionary suicide, where the evolution of the focal prey first makes the community open to the invasion of the alternative prey, which in turn makes evolutionary suicide of the focal prey possible. These new phenomena highlight the importance of studying evolution in a broader community context.Peer reviewe
Fear effect in a three-species food chain model with generalist predator
Within the framework of a food web, the foraging behavior of meso-carnivorous species is influenced by fear responses elicited by higher trophic level species, consequently diminishing the fecundity of these species. In this study, we investigate a three-species food chain model comprising of prey, an intermediate predator, and a top predator. We assume that both the birth rate and intraspecies competition of prey are impacted by fear induced by the intermediate predator. Additionally, the foraging behavior of the intermediate predator is constrained due to the presence of the top predator. It is essential to note that the top predators exhibit a generalist feeding behavior, encompassing food sources beyond the intermediate predators. The study systematically determines all feasible equilibria of the proposed model and conducts a comprehensive stability analysis of these equilibria. The investigation reveals that the system undergoes Hopf bifurcation concerning various model parameters. Notably, when other food sources significantly contribute to the growth of the top predators, the system exhibits stable behavior around the interior equilibrium. Our findings indicate that the dynamic influence of fear plays a robust role in stabilizing the system. Furthermore, a cascading effect within the system, stemming from the fear instigated by top predators, is observed and analyzed. Overall, this research sheds light on the intricate dynamics of fear-induced responses in shaping the stability and behavior of multi-species food web systems, highlighting the profound cascading effects triggered by fear mechanisms in the ecosystem
Attractors and long transients in a spatio-temporal slow-fast Bazykin's model
Spatio-temporal complexity of ecological dynamics has been a major focus of
research for a few decades. Pattern formation, chaos, regime shifts and long
transients are frequently observed in field data but specific factors and
mechanisms responsible for the complex dynamics often remain obscure. An
elementary building block of ecological population dynamics is a prey-predator
system. In spite of its apparent simplicity, it has been demonstrated that a
considerable part of ecological dynamical complexity may originate in this
elementary system. A considerable progress in understanding of the
prey-predator system's potential complexity has been made over the last few
years; however, there are yet many questions remaining. In this paper, we focus
on the effect of intraspecific competition in the predator population. In
mathematical terms, such competition can be described by an additional
quadratic term in the equation for the predator population, hence resulting in
the variant of prey-predator system that is often referred to as Bazykin's
model. We pay a particular attention to the case (often observed in real
population communities) where the inherent prey and predator timescales are
significantly different: the property known as a `slow-fast' dynamics. Using an
array of analytical methods along with numerical simulations, we provide
comprehensive investigation into the spatio-temporal dynamics of this system.
In doing that, we apply a novel approach to quantify the system solution by
calculating its norm in two different metrics such as and . We show
that the slow-fast Bazykin's system exhibits a rich spatio-temporal dynamics,
including a variety of long exotic transient regimes that can last for hundreds
and thousands of generations
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
Dynamics of a two prey and one predator system with indirect effect
none3We study a population model with two preys and one predator, considering a Holling type II functional response for the interaction between first prey and predator and taking into account indirect effect of predation. We perform the stability analysis of equilibria and study the possibility of Hopf bifurcation. We also include a detailed discussion on the problem of persistence. Several numerical simulations are provided in order to illustrate the theoretical results of the paper.openColucci R.; Diz-Pita E.; Otero-Espinar M.V.Colucci, R.; Diz-Pita, E.; Otero-Espinar, M. V
Dynamics of a two prey and one predator system with indirect effect
Altres ajuts: ConsellerÃa de Educación, Universidade e Formación Profesional (Xunta de Galicia), grant ED431C 2019/10We study a population model with two preys and one predator, considering a Holling type II functional response for the interaction between first prey and predator and taking into account indirect effect of predation. We perform the stability analysis of equilibria and study the possibility of Hopf bifurcation. We also include a detailed discussion on the problem of persistence. Several numerical simulations are provided in order to illustrate the theoretical results of the paper
Discontinuous harvesting policy in a Filippov system involving prey refuge
In this article, a non-smooth predator-prey dynamical system is considered.
Here, we discuss about sustainable harvesting in a Filippov predator-prey
system, which can produce yield and at the same time prevent over-exploitation
of bioresources. The local and global stability analysis of the two subsystems,
with and without harvesting, are studied. Furthermore, for the Filippov system,
we have performed bifurcation analysis for several key parameters like
predation rate, threshold quantity and prey refuge. Some local sliding
bifurcations are also observed for the system. Numerical simulations are
presented to illustrate the dynamical behaviour of the system