Introduction to Mathematical Modeling and Chaotic Dynamics

Predator interference effects on biological control: The “paradox” of the generalist predator revisited

“…Comparing this with empirical/statistical evidences from 19 predator-prey systems, Skalski and Gilliam [44] pointed out that the predator dependent functional responses (Hassell-Varley type [45], Beddington-DeAngelis type [46,47,48,49,50,51] and Crowley-Martin type [52]) could provide a better description of predator feeding, over a range of predator-prey abundances, and in some cases the BeddingtonDeAngelis (BD) type functional response performed even better [53]. Upadhyay and Iyengar [54] have pointed out that if predators do not waste time, interacting with one another, or if the attacks are always successful and instantaneous, then the response changes into Holling type II functional response, and the predators benefit from co-feeding.…”

Wave of chaos in a spatial eco-epidemiological system: Generating realistic patterns of patchiness in rabbit–lynx dynamics

“…Rabbits consistently account for more than 80% of the consumed biomass in the lynx diet and hence it shows that lynx is a rabbit specialist by necessity [19]. A specialist predator dies out when its favorite food is absent or is in short supply while the generalist predator switches to an alternate food option as when it faces difficulty to find its favorite preys [62]. Therefore, the specialist predator is more prone to extinction as compared to the generalist predator.…”

“…It is well known that the Kolmogorov theorem is applicable to a two component dynamical system and guarantees the existence of either a stable equilibrium point or a stable limit cycle behaviour in the positive quadrant of the phase space of the system provided certain conditions are satisfied [62]. The amplitude and the period of the stable limit cycle oscillation depend on the values of the system parameters and strongly suggest that those natural systems which seem to exhibit a persistent pattern of reasonably regular oscillations possess a stable limit cycle [36].…”

Controlling Chaotic Dynamics of a Continuous Ecological Model

“…Ayala [10] proved experimentally that two species of Drosophila could coexist upon a single limit resource. Ayala's experiment leads to a large body of work, which continuous to this day, see [1,2,6,8,[15][16][17][19][20][21][22][23][24][25]. Upadhyay [20] studied Rosenzweig-McArthur [18] model to investigate the existence of complex dynamical behavior in a food chain model involving Holling type IV functional response of the form wx d+x+x 2 /i and the effects of the intraspecific competition in the first level of the model is also studied.…”

“…Upadhyay [20] studied Rosenzweig-McArthur [18] model to investigate the existence of complex dynamical behavior in a food chain model involving Holling type IV functional response of the form wx d+x+x 2 /i and the effects of the intraspecific competition in the first level of the model is also studied. Sokol and Howell [28] suggested a simplified Holling type IV function of the form wx d+x 2 and found that it is simpler and better than the original function of Holling type IV, about how these functional responses obtained see [7,13,21,22] and [26][27][28]. Lyaponuv exponents and bifurcation diagrams are tools have been used to study the complex dynamics of the food chain models, see [3,11,12].…”