Complete Global and Bifurcation Analysis of a Stoichiometric Predator–Prey Model

Somatic Growth Dilution of a toxicant in a predator–prey model under stoichiometric constraints

“…Figure 5 uses the light level K as the bifurcation parameter. These bifurcations are similar to those obtained in the LKE model (Loladze et al, 2000) and globally investigated by Van Voorn et al (2010), Li et al (2011), and Xie et al (2016). When light levels are very low there is not enough energy in the system to support the predator population.…”

“…Data generated using XPP-AUTO. Population dynamics are similar to those obtained in Loladze et al (2000) and globally investigated in Van Voorn et al (2010), Li et al (2011), and Xie et al (2016). Figure 6 uses the toxicant contamination level T as the bifurcation parameter and a fixed light level K = 0.6 mg C/L.…”

Noise-Induced Transitions in a Nonsmooth Producer–Grazer Model with Stoichiometric Constraints

“…In this section, in order to formulate our stochastic model, we first recall the main results of model (1.1) with Holling-type II functional response from Xie et al (2018). As in Xie et al (2018), we always assume that in this paper…”

“…We refer the readers to Fig. 2 in Xie et al (2018), where the grazer nullcline (red curves) in the forward invariant region Ω is the positive x-axis and a polygonal line consisted of two line segments: x = x * , y ∈ [0, p − x * ] (denoted by l 1 ) and dx +cey = cep −ad, x ∈ [x * , min{K , cep d −a}] (denoted by l 2 ), where x * := ad ce−d ; the producer nullcline (blue curves) in the forward invariant region Ω is the positive y-axis and a parabola arc:…”

“…Model (1.1) has complex dynamics such as multiple positive equilibria and bistability (Loladze et al 2000). Later, Li et al (2011) provided a rigorous mathematical analysis for global stability results of all equilibria and the existence of limit cycles with Holling-type functional responses and fixed parameters except K , and Xie et al (2018) presented complete global and bifurcation analyses for model (1.1) with Holling type II functional response with all flexible parameters. They found that the model has four types of bistability: between an internal equilibrium and a limit cycle, between an internal equilibrium and a boundary equilibrium, between two internal equilibria, and between a boundary equilibrium and a limit cycle.…”