Predator-prey system

Brizzi, R., Delfino, G. and Pellegrini, R. (2002) Specialized mucous glands and their possible adaptive role in the males of some species of Rana (Amphibia, Anura). J. Morph. 254, 328-341. Chen, C. and Osuch, M. V. (1969) Biosynthesis of bufadienolides - 3Bhydroxycholonates as precursors in Bufo marinus bufadienolides synthesis. Biochem. Pharmacol. 18, 1797-1802. Chivers, D. P. and Smith, R. J. F. (1998) Chemical alarm signalling in aquatic predator-prey systems a review and prospectus. Ecosci. 5, 338-352. 

Until the 1950s, the rare periodic phenomena known in chemistry, such as the reaction of Bray [1], represented laboratory curiosities. Some oscillatory reactions were also known in electrochemistry. The link was made between the cardiac rhythm and electrical oscillators [2]. New examples of oscillatory chemical reactions were later discovered [3, 4]. From a theoretical point of view, the first kinetic model for oscillatory reactions was analyzed by Lotka [5], while similar equations were proposed soon after by Volterra [6] to account for oscillations in predator-prey systems in ecology. The next important advance on biological oscillations came from the experimental and theoretical studies of Hodgkin and Huxley [7], which clarified the physicochemical bases of the action potential in electrically excitable cells. The theory that they developed was later applied [8] to account for sustained oscillations of the membrane potential in these cells. Remarkably, the classic study by Hodgkin and Huxley appeared in the same year as Turing s pioneering analysis of spatial patterns in chemical systems [9]. 

Chemical alarm signaling in aquatic predator-prey systems a review and prospectus. Ecoscience 5,338-352. 

Cul] J. M. Cushing (1977), Periodic time-dependent predator prey systems, SIAM Journal on Applied Mathematics 23 972-9. 

Beddington JR, Mills CA, Beards F, et al. 1989. Long-term changes in strontium-90 concentrations within a freshwater predator-prey system. J Fish Biol 35 679-686. 

The solutions of (3.68)-(3.69) for positive parameters and generic initial conditions are oscillations, of amplitude fixed by the initial conditions, around the fixed point Z — a /a2, P = b2/b. The equation of this family of closed trajectories is ai In Z + b2 In P — biP — a2Z = constant. The oscillations are suggestive of the population oscillations observed in some real predator-prey systems, but they suffer from an important drawback the existence of the continuous family of oscillating trajectories is structurally unstable systems similar to (3.68)-(3.69) but with small additional terms either lack the oscillations, or a single limit cycle is selected out of the continuum. Thus, the model (3.68)-(3.69) can not be considered a robust model of biological interactions, which are never known with enough... 

Oscillations can also arise from the nonlinear interactions present in population dynamics (e.g. predator-prey systems). Mixing in this context is relevant for oceanic plankton populations. Phytoplankton-zooplankton (PZ) and other more complicated plankton population models often exhibit oscillatory solutions (see e.g. Edwards and Yool (2000)). Huisman and Weissing (1999) have shown that oscillations and chaotic fluctuations generated by the plankton population dynamics can provide a mechanism for the coexistence of the huge number of plankton species competing for only a few key resources (the plankton paradox ). In this chapter we review theoretical, numerical and experimental work on unsteady (mainly oscillatory) systems in the presence of mixing and stirring. 

Historically the mathematical description of predator-prey systems goes back to the times of the first World War. The Italian biologist Umberto D Ancona made the observation that in the years after the war the proportion of predatory fishes, which were caught in the Upper Adriatic Sea,... 

Gilpin, M.E. Enriched predator-prey systems Theoretical stability. Science 177, 902-904 (1972). 

Figure 3 must be regarded almost entirely as speculation, since, so far as I know, a complete figure like It has never been established for any pair of microorganisms. Salt (68) gave data for the predator-prey system Woodruffla metabollca-Parameclum aurella, but these are of limited extent and do not show the asymptotes of Figure 3. Bader (59) gave a figure for con-...

In addition to showing that X changed with respect to H/P, the results indicated also that it depended explictly on time, so that the differential equations describing the predator-prey system are probably non-autonomous. 

Fig. 4.5 Bidirectional communication in predator-prey systems mediated by chemosensation. Foraging crabs locate bivalves by homing in on the metabolites contained in the plume created by the excurrent flow (top). Chemical cues are released by foraging crabs, as well as by injured bivalves (middle). These cues suppress the response of other bivalves, and which makes them undetectable to downstream predators (bottom). Drawing courtesy to Jorge Andres Varela Ramos...

FIGURE 6.20.3 One-hundred-year record of population cycles of the snowshoe hare Lepus americanus) and the Canada lynx (Lynx canadensis), based on pelt records of the Hudson s Bay Company in Canada. Lack of anticipation in predator-prey systems lead to unstable population oscillations. (From Gotelli, N.J., A Primer of Ecology, Sinauer Associates, Sunderland, MA, 1998. With permission.)... 

Omnivores are extremely eommon in animal kingdom. but onmivoiy as a food behavior system is not yet properly understood,espeeially in a predator-prey system. Omnivory stabilizes natural systems especially... 

Chivers, D. P., and Smith, R. J. F., 1998, Chemical alarm signaling in aquatic predator-prey systems a review and prospectus, Ecoscience 5 338-352. 

Timm, U., Okubo, A. Diffusion-driven instability in a predator-prey system with time-varying diffusivities. J. Math. Biol. 30(3), 307-320(1992). http //dx.doi.org/10.1007/ BF00176153... 

Figure 5.61. Predicted behavior of a predator-prey system in single-stage CSTR culture with respect to mean residence time F and concentration of limiting substrate Sin for bacterial prey in the influent. A modification is represented by incorporating a multiple saturation predator rate of Equ. 5.206 (after lost et al, 1973, and Tsuchiya et al., 1972). The curves 1-5 separate the following regions of the operating diagram (a) total washout, (b) X2 washed out, (c) no oscillations, (d) damped oscillations, (Cj) periodic oscillations, and (Cn) sustained oscillations.

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