Unstable the system therefore exhibits bistability

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unstable. The system therefore exhibits bistability (recall that E1 is always stable). With a decrease in A, E (2) 2 becomes stable, so that for the parameters in Domain 2 the system exhibits tristability. With a further decrease in A,i.e. in Domain 1, E (2) 2 disappears in a saddle-node bifurcation (see Section 3.1). For approximately the same value of A, the coexistence state disappears as well so that, for sufficiently small values of A, the only attractor of thesystem is the extinction state. With an increase in A, i.e. in Domain 4, E (2) 2 remains unstable (a saddle point) and E3 looses its stability to become an unstable focus. For these parameters, the unstable state E3 is surrounded bya stable limit cycle that appears through the Hopf bifurcation when crossing from Domain 3 to Domain 4. With a further increase in A, the limit cycle disappears through a nonlocal bifurcation when crossing from Domain 4 toDomain 5. For parameter values from Domain 5, the only attractor of the system is the extinction state E1. From the ecological point of view, the structure of (A, c1) parameter plane means that the system is viable (in particular,ensuring sustainable production of oxygen) only in the intermediate range of A. As we will show below, this observation have important implications in the context of the climate change. 3.3 Numerical simulations In the nonspatialsystem (19-21), the information about the steady states existence and stability provides an exhausting overview of the large-time dynamics of the system. However, it does not always give enough details about the transient stageof the dynamics 12 given by the evolution of specific initial conditions. Meanwhile, there is a growing understanding that the transient dynamics may be more relevant to the dynamics of real ecosystems than the large-timeasymptotics, cf. [34, 35]. In order to make an insight into this issue, in this section we present the results of numerical simulations of the system (19-21) choosing parameter values representative of all interesting dynamicalregimes. We fix most of the parameters at some hypothetical values as B = 1.8, σ = 0.1, c1 = 0.7, c2 = 1, c3 = 1, c4 = 1, η = 0.7, δ = 1, ν = 0.01, µ = 0.1 and h = 0.1, but vary A in a broad range. Figure 5 shows the oxygen andplankton densities versus time obtained for parameter A chosen in Domain 4 or Domain 5 (cf. Fig. 4). For parameters of Fig. 5a, the system possesses a stable limit cycle so that the initial conditions promptly converge tooscillatory dynamics. Parameters of Fig. 5b are further away from the Hopf bifurcation curve (i.e. the boundary between Domains 3 and 4) so that the limit cycle is of a bigger size, which results in sustainable oscillations of amuch larger amplitude. The limit cycle disappears for parameters of Fig. 5c, so that the species densities go to extinction after just a few oscillations. Parameters of Fig. 5d are further into Domain 5, there is no limit cycle and it is

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Term
Spring
Professor
hadassah yisrael
Tags
Biodiversity, Centers for Disease Control and Prevention, Biological warfare

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