Maximizing a psychological uplift in love dynamics

In this paper, we investigate the dynamical properties of a psychological uplift in lovers. We first evaluate extensively the dynamical equations which were recently given by Rinaldi et. al., Physica A 392, pp.3231-3239 (2013). Then, the dependences of the equations on several parameters are numerically examined. From the view point of lasting partnership for lovers, especially, for married couples, one should optimize the parameters appearing in the dynamical equations to maintain the love for their respective partners. To achieve this optimization, we propose a new idea where the parameters are stochastic variables and the parameters in the next time step are given as expectations over a Boltzmann-Gibbs distribution at a finite temperature. This idea is very general and might be applicable to other models dealing with human relationships.Comment: 12 pages, 4 figures. To appear in Eds. R. Lopez-Ruiz, D. Fournier-Prunaret, Y. Nishio, C. Gracio, Nonlinear Maps and their Applications, Springer Proceedings in Mathematics & Statistic

Analytical detection of stationary and dynamic patterns in a prey-predator model with reproductive Allee effect in prey growth

Allee effect in population dynamics has a major impact in suppressing the paradox of enrichment through global bifurcation, and it can generate highly complex dynamics. The influence of the reproductive Allee effect, incorporated in the prey's growth rate of a prey-predator model with Beddington-DeAngelis functional response, is investigated here. Preliminary local and global bifurcations are identified of the temporal model. Existence and non-existence of heterogeneous steady-state solutions of the spatio-temporal system are established for suitable ranges of parameter values. The spatio-temporal model satisfies Turing instability conditions, but numerical investigation reveals that the heterogeneous patterns corresponding to unstable Turing eigen modes acts as a transitory pattern. Inclusion of the reproductive Allee effect in the prey population has a destabilising effect on the coexistence equilibrium. For a range of parameter values, various branches of stationary solutions including mode-dependent Turing solutions and localized pattern solutions are identified using numerical bifurcation technique. The model is also capable to produce some complex dynamic patterns such as travelling wave, moving pulse solution, and spatio-temporal chaos for certain range of parameters and diffusivity along with appropriate choice of initial conditions Judicious choices of parametrization for the Beddington-DeAngelis functional response help us to infer about the resulting patterns for similar prey-predator models with Holling type-II functional response and ratio-dependent functional response

Immuno-epidemiological model of two-stage epidemic growth

Epidemiological data on seasonal influenza show that the growth rate of the number of infected individuals can increase passing from one exponential growth rate to another one with a larger exponent. Such behavior is not described by conventional epidemiological models. In this work an immuno-epidemiological model is proposed in order to describe this two-stage growth. It takes into account that the growth in the number of infected individuals increases the initial viral load and provides a passage from the first stage of epidemic where only people with weak immune response are infected to the second stage where people with strong immune response are also infected. This scenario may be viewed as an increase of the effective number of susceptible increasing the effective growth rate of infected.Comment: 12 pages, 6 figure

An age-Dependent Immuno-Epidemiological Model With Distributed Recovery and Death Rates

The work is devoted to a new immuno-epidemiological model with distributed recovery and death rates considered as functions of time after the infection onset. Disease transmission rate depends on the intra-subject viral load determined from the immunological submodel. The age-dependent model includes the viral load, recovery and death rates as functions of age considered as a continuous variable. Equations for susceptible, infected, recovered and dead compartments are expressed in terms of the number of newly infected cases. The analysis of the model includes the proof of the existence and uniqueness of solution. Furthermore, it is shown how the model can be reduced to age-dependent SIR or delay model under certain assumptions on recovery and death distributions. Basic reproduction number and final size of epidemic are determined for the reduced models. The model is validated with a COVID-19 case data. Modelling results show that proportion of young age groups can influence the epidemic progression since disease transmission rate for them is higher than for other age groups

Structural sensitivity of chaotic dynamics in Hastings-Powell's model

The classical Hastings-Powell model is well known to exhibit chaotic dynamics in a three-species food chain. Chaotic dynamics appear through period-doubling bifurcation of stable coexistence limit cycle around an unstable interior equilibrium point. A specific choice of parameter value leads to a situation where the chaotic attractor disappears through a collision with an unstable limit cycle. As a result, the top predator goes to extinction. Here we explore the structural sensitivity of this phenomenon by replacing the Holling type II functional responses with Ivlev functional responses. Here we prove the existence of two Hopf-bifurcation thresholds and numerically detect the existence of an unstable limit cycle. The model with Ivlev functional responses does not indicate any possibility of extinction of the top predator. Further, the choice of functional responses depicts a significantly different picture of the coexistence of the three species involved with the model

An Epidemic Model With Time Delays Determined By the infectivity and Disease Durations

We propose an epidemiological model with distributed recovery and death rates. It represents an integrodifferential system of equations for susceptible, exposed, infectious, recovered and dead compartments. This model can be reduced to the conventional ODE model under the assumption that recovery and death rates are uniformly distributed in time during disease duration. Another limiting case, where recovery and death rates are given by the delta-function, leads to a new point-wise delay model with two time delays corresponding to the infectivity period and disease duration. Existence and positiveness of solutions for the distributed delay model and point-wise delay model are proved. The basic reproduction number and the final size of the epidemic are determined. Both, the ODE model and the delay models are used to describe COVID-19 epidemic progression. The delay model gives a better approximation of the Omicron data than the conventional ODE model from the point of view of parameter estimation

Attractors and long transients in a spatio-temporal slow-fast Bazykin's model

Spatio-temporal complexity of ecological dynamics has been a major focus of research for a few decades. Pattern formation, chaos, regime shifts and long transients are frequently observed in field data but specific factors and mechanisms responsible for the complex dynamics often remain obscure. An elementary building block of ecological population dynamics is a prey-predator system. In spite of its apparent simplicity, it has been demonstrated that a considerable part of ecological dynamical complexity may originate in this elementary system. A considerable progress in understanding of the prey-predator system's potential complexity has been made over the last few years; however, there are yet many questions remaining. In this paper, we focus on the effect of intraspecific competition in the predator population. In mathematical terms, such competition can be described by an additional quadratic term in the equation for the predator population, hence resulting in the variant of prey-predator system that is often referred to as Bazykin's model. We pay a particular attention to the case (often observed in real population communities) where the inherent prey and predator timescales are significantly different: the property known as a `slow-fast' dynamics. Using an array of analytical methods along with numerical simulations, we provide comprehensive investigation into the spatio-temporal dynamics of this system. In doing that, we apply a novel approach to quantify the system solution by calculating its norm in two different metrics such as $C^0$ and $L^2$. We show that the slow-fast Bazykin's system exhibits a rich spatio-temporal dynamics, including a variety of long exotic transient regimes that can last for hundreds and thousands of generations

Normal form for singular Bautin bifurcation in a slow-fast system with Holling type III functional response

Over the last few decades, complex oscillations of slow-fast systems have been a key area of research. In the theory of slow-fast systems, the location of singular Hopf bifurcation and maximal canard is determined by computing the first Lyapunov coefficient. In particular, the analysis of canards is based on the genericity condition that the first Lyapunov coefficient must be non-zero. This manuscript aims to further extend the results to the case where the first Lyapunov coefficient vanishes. For that, the analytic expression of the second Lyapunov coefficient and the investigation of the normal form for codimension-2 singular Bautin bifurcation in a predator-prey system is done by explicitly identifying the locally invertible parameter-dependent transformations. A planar slow-fast predator-prey model with Holling type III functional response is considered here, where the prey population growth is affected by the weak Allee effect, and the prey reproduces much faster than the predator. Using geometric singular perturbation theory, normal form theory of slow-fast systems, and blow-up technique, we provide a detailed mathematical investigation of the system to show a variety of rich and complex nonlinear dynamics including but not limited to the existence of canards, relaxation oscillations, canard phenomena, singular Hopf bifurcation, and singular Bautin bifurcation. Additionally, numerical simulations are conducted to support the theoretical findings