arXiv subjects

Susmita Sadhu

Publications and source records attributed to Susmita Sadhu.

6 recordsLinked to original sources

arXiv2022

Analysis of long transients and detection of early warning signals of extinction in a class of predator-prey models exhibiting bistable behavior

In this paper, we develop a method of analyzing long transient dynamics in a class of predator-prey models with two species of predators competing explicitly for their common prey, where the prey evolves on a faster timescale than the predators. In a parameter regime near a {\em{singular zero-Hopf bifurcation}} of the coexistence equilibrium state, we assume that the system under study exhibits bistability between a periodic attractor that bifurcates from the singular Hopf point and another attractor, which could be a periodic attractor or a point attractor, such that the invariant manifolds of the coexistence equilibrium point play central roles in organizing the dynamics. To find whether a solution that starts in a vicinity of the coexistence equilibrium approaches the periodic attractor or the other attractor, we reduce the equations to a suitable normal form, and examine the basin boundary near the singular Hopf point. A key component of our study includes an analysis of the long transient dynamics, characterized by their rapid oscillations with a slow variation in amplitude, by applying a moving average technique. We obtain a set of necessary and sufficient conditions on the initial values of a solution near the coexistence equilibrium to determine whether it lies in the basin of attraction of the periodic attractor. As a result of our analysis, we devise a method of identifying early warning signals, significantly in advance, of a future crisis that could lead to extinction of one of the predators. The analysis is applied to the predator-prey model considered in [\emph{Discrete and Continuous Dynamical Systems - B} 2021, 26(10), pp. 5251-5279] and we find that our theory is in good agreement with the numerical simulations carried out for this model.

math.DS↗

arXiv2022

Complex oscillatory patterns in a three-timescale model of a generalist predator and a specialist predator competing for a common prey

In this paper, we develop and analyze a model that studies the interaction between a specialist predator, a generalist predator, and their common prey in a two-trophic ecosystem featuring three timescales. We assume that the prey operates on a faster timescale, while the specialist and generalist predators operate on slow and superslow timescales respectively. Treating the predation efficiency of the generalist predator as the primary varying parameter and the proportion of its diet formed by the prey species under study as the secondary parameter, we obtain a host of rich and interesting dynamics, including relaxation oscillations, mixed-mode oscillations (MMOs), subcritical elliptic bursting patterns, torus canards, and mixed-type torus canards. By grouping the timescales into two classes and using the timescale separation between classes, we apply one-fast/two-slow and two-fast/one-slow analysis techniques to gain insights about the dynamics. Using the geometric properties and flows of the singular subsystems, in combination with bifurcation analysis and numerical continuation of the full system, we classify the oscillatory dynamics and discuss the transitions from one type of dynamics to the other. The types of oscillatory patterns observed in this model are novel in population models featuring three-timescales; some of which qualitatively resemble natural cycles in small mammals and insects. Furthermore, oscillatory dynamics displaying torus canards, mixed-type torus canards, and MMOs experiencing a delayed loss of stability near one of the invariant sheets of the self-intersecting critical manifold before getting attracted to the adjacent attracting sheet of the critical manifold have not been previously reported in three-timescale models.

math.DS↗

arXiv2021

Analysis of long-term transients and detection of early warning signs of major population changes in a two-timescale ecosystem

Identifying early warning signs of sudden population changes and mechanisms leading to regime shifts are highly desirable in population biology. In this paper, a two-trophic ecosystem comprising of two species of predators, competing for their common prey, with explicit interference competition is considered. With proper rescaling, the model is portrayed as a singularly perturbed system with fast prey dynamics and slow dynamics of the predators. In a parameter regime near singular Hopf bifurcation, chaotic mixed-mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations are observed as long-lasting transients before the system approaches its asymptotic state. To analyze the dynamical cause that initiates a large amplitude oscillation in an MMO orbit, the model is reduced to a suitable normal form near the singular-Hopf point. The normal form possesses a separatrix surface that separates two different types of oscillations. A large amplitude oscillation is initiated if a trajectory moves from the "inner" to the "outer side" of this surface. A set of conditions on the normal form variables are obtained to determine whether a trajectory would exhibit another cycle of MMO dynamics before experiencing a regime shift (i.e. approaching its asymptotic state). These conditions can serve as early warning signs for a sudden population shift in an ecosystem.

q-bio.PE↗

arXiv2020

Complex oscillatory patterns near singular Hopf bifurcation in a two time-scale ecosystem

We consider an ecological model consisting of two species of predators competing for their common prey with explicit interference competition. With a proper rescaling, the model is portrayed as a singularly perturbed system with one-fast (prey dynamics) and two-slow variables (dynamics of the predators). The model exhibits variety of rich and interesting dynamics, including, but not limited to mixed mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations, relaxation oscillations and bistability between a semi-trivial equilibrium state and a coexistence oscillatory state. Existence of co-dimenison two bifurcations such as fold-Hopf and generalized Hopf bifurcations make the system further intriguing. More interestingly, in a neighborhood of {\emph{singular Hopf}} bifurcation, long lasting transient dynamics in form of chaotic MMOs or relaxation oscillations are observed as the system approaches the periodic attractor born out of supercritical Hopf bifurcation or a semi-trivial equilibrium state respectively. The transient dynamics could persist for hundreds or thousands of generations before the ecosystem experiences a regime shift. The time series of population cycles with different types of irregular oscillations arising in this model stem from a biological realistic feature, namely, by the variation in the intraspecific competition amongst the predators. To explain these oscillations, we use bifurcation analysis and methods from {\emph{geometric singular perturbation theory}}.

math.DS↗

arXiv2017

Stochasticity induced mixed-mode oscillations and distribution of recurrent outbreaks in an ecosystem

The effect of stochasticity, in the form of Gaussian white noise, in a predator-prey model with two distinct time-scales is presented. A supercritical singular Hopf bifurcation yields a Type II excitability in the deterministic model. We explore the effect of stochasticity in the excitable regime, leading to dynamics that are not anticipated by its deterministic counterpart. The stochastic model admits several kinds of noise-driven mixed-mode oscillations which capture the intermediate dynamics between two cycles of population outbreaks. Depending on the strength of noise, the prey population exhibits intermediate to high-amplitude fluctuations (related to moderate or severe outbreaks respectively). We classify these fluctuations as isolated or intermittent or as clusters depending on their recurrences. We study the distribution of the random variable $N$, representing the number of small oscillations between successive spikes, as a function of the noise intensity and the distance to the Hopf bifurcation. The distribution of $N$ is "asymptotically geometric" with the corresponding parameter related to the principal eigenvalue of a substochastic Markov chain. Finally, the stochastic model is transformed into its "normal form" which is used to obtain an estimate of the probability of repeated outbreaks.

math.DS↗

arXiv2017

Stochastic mixed-mode oscillations in a three-species predator-prey model

The effect of demographic stochasticity, in the form of Gaussian white noise, in a predator-prey model with one fast and two slow variables is studied. We derive the stochastic differential equations (SDEs) from a discrete model. For suitable parameter values, the deterministic drift part of the model admits a folded node singularity and exhibits a singular Hopf bifurcation. We focus on the parameter regime near the Hopf bifurcation, where small amplitude oscillations exist as stable dynamics in the absence of noise. In this regime, the stochastic model admits noise-driven mixed-mode oscillations (MMOs), which capture the intermediate dynamics between two cycles of population outbreaks. We perform numerical simulations to calculate the distribution of the random number of small oscillations between successive spikes for varying noise intensities and distance to the Hopf bifurcation. We also study the effect of noise on a suitable Poincaré map. Finally, we prove that the stochastic model can be transformed into a normal form near the folded node, which can be linked to recent results on the interplay between deterministic and stochastic small amplitude oscillations. The normal form can also be used to study the parameter influence on the noise level near folded singularities.

math.DS↗