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

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}}

Existence and asymptotic analysis of solutions of singularly perturbed boundary value problems

We study existence and uniform asymptotic expansions of solutions of two different classes of singularly perturbed boundary value problems. The first boundary value problem that we consider is ε y' + 2y'+ f(y) = 0, y(0) = y(A) = 0,where f is a smooth, positive increasing function satisfying certain properties and A > 0. We will show that the problem has two solutions for certain values of A. We will also derive and prove a uniform asymptotic expansion of the smaller solution when f(y) = e^y and A = 1. The second boundary value problem that we consider is ε² y' = y(q(x, ε) -y), y(-1)= α_-, y(1)=α_+,where q(x, ε) is a smooth function with uniformly bounded derivatives and is uniformly bounded from below by a positive constant q_∗ for ε sufficiently small. The boundary values α_± are specified positive numbers bounded from above by q_∗. We will derive uniform asymptotic expansion of solutions to this problem that have 3 or fewer critical points

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

In this paper, we develop a method of detecting an early warning signal of catastrophic population collapse in a class of predator-prey models with two species of predators competing for their common prey, where the prey evolves on a faster timescale than the predators. In a parameter regime near {\em{singular Hopf bifurcation}} of a coexistence equilibrium point, we assume that the class of models exhibits bistability between a periodic attractor and a boundary equilibrium point, where the invariant manifolds of the coexistence equilibrium play central roles in organizing the dynamics. To determine whether a solution that starts in a vicinity of the coexistence equilibrium approaches the periodic attractor or the point attractor, we reduce the equations to a suitable normal form, which is valid near the singular Hopf bifurcation, and study its geometric structure. A key component of our study includes an analysis of the transient dynamics, characterized by their rapid oscillations with a slow variation in amplitude, by applying a moving average technique. As a result of our analysis, we could devise a method for 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

Effect of frying on physicochemical properties of sesame and soybean oil blend

Most common cooking oil, such as soybean oil, can not be used for high-temperature applications, as they are highly susceptible to oxidation. Sesame seed oil rich in natural antioxidants provides high oxidative stability. Therefore, blending sesame oil with soybean oil offer improved oxidative stability. This study aims to determine the effect of frying on the physicochemical properties of sesame and soyabean oil blend. Soybean oil (SO) was blended with sesame seed oil (SSO) in the ratio of A-40:60, B-60:40 and C-50:50 so as to enhance its market acceptability. The changes occurring in soybean and sesame seed oil blend during repeated frying cycles were monitored. The parameters assessed were: Refractive index, specific gravity, viscosity, saponification value, free fatty acid (FFA) , peroxide value, and acid value. Fresh and fried oil blends were also characterised by Fourier Transform Infrared Spectroscopy (FTIR). No significant changes were observed for refractive index and specific gravity values in oil blends. Viscosity of blend B blend was the least, making it desirable for cooking purposes. However, FFA, acid value and peroxide value increased after each frying cycle. The increment of FFA and AV was found low for blend A (10% and 10%,) than blend B (27%,13%) and blend C (13%,13%). The peroxide value of all samples was within the acceptable range. The results of the present study definitely indicated that blending sesame oil with soybean oil could produce an oil blend which is economically feasible and provide desirable physicochemical properties for cooking purposes

The entry-exit theorem and relaxation oscillations in slow-fast planar systems

The entry-exit theorem for the phenomenon of delay of stability loss for certain types of slow-fast planar systems plays a key role in establishing existence of limit cycles that exhibit relaxation oscillations. The general existing proofs of this theorem depend on Fenichel\u27s geometric singular perturbation theory and blow-up techniques. In this work, we give a short and elementary proof of the entry-exit theorem based on a direct study of asymptotic formulas of the underlying solutions. We employ this theorem to a broad class of slow-fast planar systems to obtain existence, global uniqueness and asymptotic orbital stability of relaxation oscillations. The results are then applied to a diffusive predator-prey model with Holling type II functional response to establish periodic traveling wave solutions. Furthermore, we extend our work to another class of slow-fast systems that can have multiple orbits exhibiting relaxation oscillations, and subsequently apply the results to a two time-scale Holling-Tanner predator-prey model with Holling type IV functional response. It is generally assumed in the literature that the non-trivial equilibrium points exist uniquely in the interior of the domains bounded by the relaxation oscillations; we do not make this assumption in this paper