Predator–prey interaction with harvesting: mathematical study with biological ramifications

In this article, we study a simple predator–prey interaction where predator population is subjected to harvesting. Our qualitative analysis shows different outcomes including switching of stability, oscillations and deterministic extinction. Theoretically observed results have been tested with the parameter values of Paramecium aurelia and its predator Didinium nasutum. Study reveals that harvesting effort and predator’s attack rate may be used as control parameters for the system. Simulation results also indicate that the system may exhibit bistability for some parametric region. Our study also gives the possible answer to the question – why do we frequently observe coexisting predator–prey system in natural system? The study may be helpful to design control strategy for harvested predator–prey system.     

一类多菌株STD模型的共存性

我们已经研究过一类拥有两种菌株的异性传播的性传染病模型.得到了边界平衡点稳定的充要条件,并确认在边界平衡点的稳定性和正平衡点的存在性之间存在着很强的联系.但是只给出了特殊条件下正平衡点稳定的充要条件,这篇文章将就以前没解决的问题,对这类模型给出完整的分析.     

An angiogenesis model with nonlinear chemotactic response and flux at the tumor boundary

In this paper we consider a parabolic problem as well as its stationary counterpart of a model arising in angiogenesis. The problem includes a chemotaxis type term and a nonlinear boundary condition at the tumor boundary. We show that the parabolic problem admits a unique positive global in time solution. Moreover, by bifurcation methods, we show the existence of coexistence states and also we study the local stability of the semi-trivial states.     

Coexistence and asymptotic periodicity in a competitor-competitor-mutualist model

In this paper, the competitor-competitor-mutualist three-species Lotka-Volterra model is discussed. Firstly, by Schauder fixed point theory, the coexistence state of the strongly coupled system is given. Applying the method of upper and lower solutions and its associated monotone iterations, the true solutions are constructed. Our results show that this system possesses at least one coexistence state if cross-diffusions and cross-reactions are weak. Secondly, the existence and asymptotic behavior of T-periodic solutions for the periodic reaction-diffusion system under homogeneous Dirichlet boundary conditions are investigated. Sufficient conditions which guarantee the existence of T-periodic solution are also obtained.     

Crowding effects promote coexistence in the chemostat

This paper deals with an almost-global stability result for a particular chemostat model. It deviates from the classical chemostat because crowding effects are taken into consideration. This model can be rewritten as a negative feedback interconnection of two systems which are monotone (as input/output systems). Moreover, these subsystems behave nicely when subject to constant inputs. This allows the use of a particular small-gain theorem which has recently been developed for feedback interconnections of monotone systems. Application of this theorem requires—at least approximate—knowledge of two gain functions associated to the subsystems. It turns out that for the chemostat model proposed here, these approximations can be obtained explicitly and this leads to a sufficient condition for almost-global stability. In addition, we show that coexistence occurs in this model if the crowding effects are large enough.     

Measurements of the isochoric heat capacity,the critical point (TC, ρC) and vapor–liquid coexistence curve (TS, ρS) of high-purity toluene near the critical point

Isochoric heat capacities (C_V, V, T), phase boundary properties (T_S, ρ_S) and the critical (T_C, ρ_C) parameters for high-purity (0.9999+ mole fraction) toluene have been measured with a high temperature, high pressure, nearly constant volume adiabatic calorimeter and quasi-static thermogram technique. Measurements were made at three selected liquid and vapor isochores 777.8, 555.25, and 214.64 kg m⁻³ in the temperature range from 379 to 591 K. For five near-critical isochores 268.68, 281.68, 296.62, 301.52, and 318.28 kg m⁻³, the measurements were made in the immediate vicinity of the coexistence curve in order to accurately determine the phase transition temperatures (T_S, ρ_S) (shape of the coexistence curve near the critical point) and the critical parameters (T_C, ρ_C). The total combined uncertainty of heat capacity, density, and temperature measurements were estimated to be less than 2%, 0.06%, and 15 mK, respectively. The uncertainties reported in this paper are expanded uncertainties at the 95% confidence level with a coverage factor of k = 2. The uncertainty of the phase transition and the critical temperature value was 0.02 K. The Krichevskii parameter for some toluene-containing binary mixtures was calculated. The derived values of the Krichevskii parameter were used to estimate the effect of dilute impurities on the critical parameters of toluene. The measured values of saturated density near the critical point were interpreted in terms of the “complete scaling” theory in order to study singularity behavior of the coexistence curve diameter. The measured isochoric heat capacities and saturated densities were compared with the data reported by other authors and values calculated from an equation of state and other correlations.     

The measurements of coexistence curves and light scattering for {xC6H5CN + (1 − x)CH3(CH2)6CH3} in the critical region

The coexistence curves and light scattering data for a critical solution of (benzonitrile + octane) have been reported. The critical exponents relating to the difference in the density variables between two coexisting phases β, the osmotic compressibility γ, and the correlation length ν have been deduced and the values are consistent with the 3D-Ising value in the range close to the critical point. The experimental results of the coexistence curves have also been analyzed to examine the Wegner correction terms and the behavior of the diameter of the coexistence curves. The light scattering data are well described by the crossover model proposed by Anisimov and Sengers, and show a monotonic crossover of the critical exponents γ and ν from its 3D-Ising value to the mean-field value as the temperature departures from the critical point. Furthermore, the dependences of the critical amplitudes on the mass of n-alkane for the binary solutions of (benzonitrile + n-alkane) have been discussed.     

Dynamics in a ratio-dependent predator-prey model with predator harvesting

The objective of this paper is to study systematically the dynamical properties of a ratio-dependent predator-prey model with nonzero constant rate predator harvesting. It is shown that the model has at most two equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation), the subcritical and supercritical Hopf bifurcations. These results reveal far richer dynamics compared to the model with no harvesting and different dynamics compared to the model with nonzero constant rate prey harvesting in [D. Xiao, L. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM Appl. Math. 65 (2005) 737-753]. Biologically, it is shown that nonzero constant rate predator harvesting can prevent mutual extinction as a possible outcome of the predator prey interaction, and remove the singularity of the origin, which was regarded as “pathological behavior” for a ratio-dependent predator prey model in [P. Yodzis, Predator-prey theory and management of multispecies fisheries, Ecological Applications 4 (2004) 51-58].     

Positive steady states for a prey-predator model with some nonlinear diffusion terms

This paper discusses a prey-predator system with strongly coupled nonlinear diffusion terms. We give a sufficient condition for the existence of positive steady state solutions. Our proof is based on the bifurcation theory. Some a priori estimates for steady state solutions will play an important role in the proof.     

Coexistence phenomenon in autoparametric excitation of two degree of freedom systems

Coexistence phenomenon refers to the absence of expected tongues of instability in parametrically excited systems. In this paper we obtain sufficient conditions for coexistence to occur in the generalized Ince equation