Interspecific density-dependent model of predator–prey relationship in the chemostat

Abstract

The objective of this study is to analyze a model of competition for one resource in the chemostat with general inter-specific density-dependent growth rates, taking into account the predator-prey relationship. This relationship is characterized by the fact that the prey species promotes the growth of the predator species which in turn inhibits the growth of the first species. The model is a three-dimensional system of ordinary differential equations. With the same dilution rates, the model can be reduced to a planar system where the two models have the same local and even global behavior. The existence and stability conditions of all steady states of the reduced model in the plane are determined according to the operating parameters. Using the nullcline method, we present a geometric characterization of the existence and stability of all equilibria showing the multiplicity of coexistence steady states. The bifurcation diagrams illustrate that the steady states can appear or disappear only through saddle-node or transcritical bifurcations. Moreover, the operating diagrams describe the asymptotic behavior of this system by varying the control parameters and show the effect of the inhibition of predation on the emergence of the bistability region and the reduction until the disappearance of the coexistence region by increasing this inhibition parameter.

A Mathematical Model of Anaerobic Digestion with Syntrophic Relationship, Substrate Inhibition, and Distinct Removal Rates

“…On the other hand, our study provides an important tool for the experimentation which is the operating diagram showing the behavior of the syntrophic model (1.2) according to the control parameters D, s in 0 and s in 1 , when all biological parameters are fixed. This operating diagram is often studied numerically or theoretically both in the biological literature [36,42,51,54] and the mathematical literature [1,2,11,12,13,18,19,23,25,32,38,40,41,52,53].…”

Best Operating Conditions for Biogas Production in Some Simple Anaerobic Digestion Models

“…Its importance for bioreactors was emphasized in [36]. This diagram is often constructed both in the biological literature [15,29,[35][36][37][38] and the mathematical literature [3,21,30,[39][40][41][42][43][44].…”

“…Assume that Hypotheses 2, 3, 6, and 7 are satisfied. Then G 01 D, S in 2 reaches its maximum at D * 0 S in 2 , defined by (39) and G 11 D, S in 2 , S in 2 reaches its maximum at the right-hand end of its defining interval, or at D * 1 S in 1 , S in 2 , defined by (41).…”

“…Consider the OD for which S in 2 is equal to the fixed value considered and look for the intersections of Γ 0 and Γ 1 with the horizontal line where S in 1 is kept constant at the fixed value considered. The abscissas of these intersections are the optimal dilution rates D * 0 S in 2 and D * 1 S in 2 defined by ( 39) and (41), respectively.…”

Mortality can produce limit cycles in density-dependent models with a predator-prey relationship

“…Several suggestions to address the disparity in predictions of models involving multiple microbial species with observations from the laboratory can be found in the literature. Note these various mechanisms to promote coexistence: the intraspecific and interspecific competition [1,10], the flocculation [16,18,19,20,21], the densitydependence of the growth functions [7,17,25,30,31,32,33,36], the predator-prey interactions [6,29], the complex food webs [2,26,55], the presence of inhibitors that affects the strongest competitor [3,11,12], the commensalistic relationship [4,5,45], and the syntrophic relationship [9,15,43,56].…”