Simulation of human population dynamics by a hyperlogistic time-delay equation

Haberl, Helmut; Aubauer, H. P.

doi:10.1016/s0022-5193(05)80640-x

Cited by 6 publications

(8 citation statements)

References 4 publications

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“…This time-delay equation is capable of simulating a population that can essentially surmount the carrying capacity K for a limited period of time, then dropping below it subsequently [19].…”

Section: (X) Hyperlogistic Time-delay Equationsmentioning

confidence: 99%

Extreme events in population dynamics with functional carrying capacity

Yukalov

Yukalova

Sornette

2012

Eur. Phys. J. Spec. Top.

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A class of models is introduced describing the evolution of population species whose carrying capacities are functionals of these populations. The functional dependence of the carrying capacities reflects the fact that the correlations between populations can be realized not merely through direct interactions, as in the usual predator-prey LotkaVolterra model, but also through the influence of species on the carrying capacities of each other. This includes the self-influence of each kind of species on its own carrying capacity with delays. Several examples of such evolution equations with functional carrying capacities are analyzed. The emphasis is given on the conditions under which the solutions to the equations display extreme events, such as finite-time death and finite-time singularity. Any destructive action of populations, whether on their own carrying capacity or on the carrying capacities of co-existing species, can lead to the instability of the whole population that is revealed in the form of the appearance of extreme events, finite-time extinctions or booms followed by crashes. PACS: 02.30.Hq, 02.30.Ks, 87.10.Ed, 87.23.Ce, 87.23.Cc, 87.23.Ge, 87.23.Kg, 89.65.Gh Keywords: Population evolution, Functional carrying capacity, Punctuated solution, Models of symbiosis, Nonlinear differential equations, Extreme events, Finite-time death, Finite-time singularity, Evolutional booms and crashes 1 1 Brief survey of population models Evolution equations, describing population dynamics, are widely employed in various branches of biology, ecology, and sociology. The main forms of such equations are given by the variants of the predator-prey Lotka-Volterra models. In this paper, we introduce a novel class of models whose principal feature, making them different from other models, is the functional dependence of the population carrying capacities on the population species. This general class of models allows for different particular realizations characterizing specific correlations between coexisting species. The functional dependence of the carrying capacities describes the mutual influence of species on the carrying capacities of each other, including the self-influence of each kind of species on its own capacity. Such a dependence is, both mathematically and biologically, principally different from the direct interactions typical of the predator-prey models. Before formulating the general approach, we give in this section a brief survey of the main known models of population dynamics. This will allow us to stress the basic difference of our approach from other models used for describing the population dynamics in biology, ecology, and sociology. (i) Predator-prey Lotka-Volterra modelThe first model, describing interacting species, one of which is a predator with population N 1 , and another is a prey with population N 2 , has been the Lotka-Volterra [1, 2] modelwhere all coefficients are positive numbers. It is easy to show that the solutions to these equations are bound oscillating functions of time.(...

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Section: (X) Hyperlogistic Time-delay Equationsmentioning

confidence: 99%

Extreme events in population dynamics with functional carrying capacity

Yukalov

Yukalova

Sornette

2012

Eur. Phys. J. Spec. Top.

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“…From the solution of equation (1) follows that the doubling time of the initial population (t d ) is related with the growth rate by t d = ln 2/r. For example, with the data of the world population, [4], we can determine the variation of the doubling time or the growth rate of the human population along historical times, Fig. 1.…”

Section: One Species Interaction With the Environmentmentioning

confidence: 99%

“…For the human population, the agreement is not so good, being dependent on technological developments, sociological trends and other factors, [2]. Depending on the data set, and from country to country, some authors find a good fit between the solutions of the logistic equation and demography data (see for example [4]), and others propose empirical models based on the delayed logistic equation, x (t) = rx α (1−x(t−T )/K), [4].…”

Section: One Species Interaction With the Environmentmentioning

confidence: 99%

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Mathematical Models in Population Dynamics and Ecology

Dilão

2006

Biomathematics

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We introduce the most common quantitative approaches to population dynamics and ecology, emphasizing the different theoretical foundations and assumptions. These populations can be aggregates of cells, simple unicellular organisms, plants or animals.The basic types of biological interactions are analysed: consumer-resource, prey-predation, competition and mutualism. Some of the modern developments associated with the concepts of chaos, quasi-periodicity, and structural stability are discussed. To describe short-and long-range population dispersal, the integral equation approach is derived, and some of its consequences are analysed. We derive the standard McKendrick age-structured density dependent model, and a particular solution of the McKendrick equation is obtained by elementary methods. The existence of demography growth cycles is discussed, and the differences between mitotic and sexual reproduction types are analysed.

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“…Such equations can be classified into three main classes, depending on the nature of the dynamics of the carrying capacity. The first class, independently of whether the growth rate is a nonlinear function without or with delay of the population, assumes that the carrying capacity is a constant quantity given once for all that describes the total resources available to the population, in agreement with the initial understanding of the carrying capacity (e.g., [Haberl & Aubauer, 1992;Varfolomeyev & Gurevich,…”

Section: Introductionmentioning

confidence: 99%

Population Dynamics with Nonlinear Delayed Carrying Capacity

Yukalov

Yukalova

Sornette

2014

Int. J. Bifurcation Chaos

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We consider a class of evolution equations describing population dynamics in the presence of a carrying capacity depending on the population with delay. In an earlier work, we presented an exhaustive classification of the logistic equation where the carrying capacity is linearly dependent on the population with a time delay, which we refer to as the "linear delayed carrying capacity" model. Here, we generalize it to the case of a nonlinear delayed carrying capacity. The nonlinear functional form of the carrying capacity characterizes the delayed feedback of the evolving population on the capacity of their surrounding by either creating additional means for survival or destroying the available resources. The previously studied linear approximation for the capacity assumed weak feedback, while the nonlinear form is applicable to arbitrarily strong feedback. The nonlinearity essentially changes the behavior of solutions to the evolution equation, as compared to the linear case. All admissible dynamical regimes are analyzed, which can be of the following types: punctuated unbounded growth, punctuated increase or punctuated degradation to a stationary state, convergence to a stationary state with sharp reversals of plateaus, oscillatory attenuation, everlasting fluctuations, everlasting up-down plateau reversals, and divergence in finite time. The theorem is proved that, for the case characterizing the evolution under gain and competition, solutions are always bounded, if the feedback is destructive. We find that even a small noise level profoundly affects the position of the finite-time singularities. Finally, we demonstrate the feasibility of predicting the critical time of solutions having finite-time singularities from the knowledge of a simple quadratic approximation of the early

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Simulation of human population dynamics by a hyperlogistic time-delay equation

Cited by 6 publications

References 4 publications

Extreme events in population dynamics with functional carrying capacity

Extreme events in population dynamics with functional carrying capacity

Mathematical Models in Population Dynamics and Ecology

Population Dynamics with Nonlinear Delayed Carrying Capacity

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