Source: https://www.groundai.com/project/population-dynamics-with-nonlinear-delayed-carrying-capacity/
Timestamp: 2019-04-19 23:10:28+00:00

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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 time dynamics.
There exists a number of evolution equations characterizing population dynamics, which have been applied to numerous concrete systems, ranging from populations of human and biological species to the development of firms and banks (see review articles [Kapitza(1996), Hern(1999), Korotayev(2007), Yukalov et al.(2012a)] ). The mathematical structure of these equations usually represents some generalizations of the logistic equation. 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(2001), Hui & Chen(2005), Gabriel et al.(2005), Berezowski & Fudala(2006), Arino et al.(2006)] ). The second class allows the carrying capacity to change as a function of time, but for exogenous reasons, either by explicitly prescribing its evolution or by specifying its own independent dynamics for instance also given by a logistic equation [Dolgonosov & Naidenov(2006), Pongvuthithum & Likasiri(2010)] .
The third class of equations interprets the carrying capacity as a functional of the population itself, implying that the population does influence the carrying capacity, either by producing additional means for survival or by destroying the available resources [Yukalov et al.(2009), Yukalov et al.(2012a), Yukalov et al.(2012b)] . This feedback makes it possible to describe the regime of punctuated evolution, which is often observed in a variety of biological, social, economic, and financial systems [Yukalov et al.(2009), Yukalov et al.(2012a)] .
In our previous articles [Yukalov et al.(2009), Yukalov et al.(2012a)] , the carrying capacity was approximated by a linear dependence on the population variable, which, strictly speaking, assumes that the population influence should be smaller than the initial given capacity. In the present paper, we generalize the approach by accepting a nonlinear carrying capacity that allows us to consider a population influence of arbitrary strength. Moreover, the population variable enters the carrying capacity with a time delay, since to either create or destroy resources requires time.
The outline of the present paper is as follows. In Sec. 2, we explain the deficiency that is typical of the linear capacity specification and suggest the way of generalizing its form to a nonlinear expression, by the use of the theory of self-similar approximations. The existence and general conditions for the stability of evolutionary stationary states are formulated in Sec. 3. The temporal behavior of solutions essentially depends on the system parameters characterizing different prevailing situations, when the main features are described as gain and competition (Sec. 4), loss and cooperation (Sec. 5), loss and competition or gain and cooperation (Sec. 6). The equations display a rich variety of dynamical regimes, including punctuated unbounded growth, punctuated increase or punctuated degradation to a stationary state, convergence to a stationary state with sharp reversals of plateaus, oscillatory convergence, everlasting fluctuations, everlasting up-down plateau reversals, and divergence in finite time. All admissible dynamical regimes are studied and illustrated. The role of noise on the dynamics is investigated in Sec. 7. The possibility of predicting finite-time singularities by observing only the initial stage of motion is discussed in Sec. 8. Section 9 concludes.
Here and in what follows, we use the dimensionless variable for the population x(t) as a function of time t. The reduction of the dimensional equation to the dimensionless form has been explained in full details in our previous papers [Yukalov et al.(2009), Yukalov et al.(2012a)] and we do not repeat it here.
depends on the population at an earlier time, with a constant delay time τ, which embodies that any influence of the population on the capacity requires time in order to either create additional means or to destroy the given resources. Here and in what follows, we use the term population, although the variable x can characterize either population, or firm assets, or other financial and economic indices [Yukalov et al.(2009), Yukalov et al.(2012a)] .
Various admissible interpretations of the equation and possible applications have been described in the published papers [Yukalov et al.(2009), Yukalov et al.(2012a)] .
with the parameter b1 describing either destructing action of population on the resources, when b1<0, or creative population activity, if b1>0.
Being in the denominator of the second term of (1), the vanishing capacity leads to the appearance of divergent or non-smooth solutions. In some cases, having to do with financial and economic applications, the arising negative capacity can be associated with the leverage effect [Yukalov et al.(2012a)] . However, in the usual situation, the solution divergencies, caused by the zero denominator, look rather unrealistic, reminding of mathematical artifacts. Therefore, it would be desirable to define a carrying capacity that would not cross zero in finite time.
It would be possible to replace the linear form (4) by some nonlinear function. This, however, would be a too arbitrary and ambiguous procedure, since it would be always unclear why this or that particular function has been chosen. In order to justify the choice of a nonlinear function, we propose the following procedure to select the form of the nonlinear carrying capacity.
where different terms describe the influence with different action intensity. Since expression (4) can be interpreted as the first-order term in the general series expansion (5) with, in principle, infinite many terms, it is convenient to think of it as the general expansion of some nonlinear function to be determined by a suitable summation. We thus propose to construct a nonlinear extension of (4) by defining an effective sum of these series (5). A standard way to realize the summation (5) is via Padé approximants [Baker & Graves-Moris(1996)] . However, as is well known, the Padé approximants are very often plagued by the occurrence of artificial zeroes and divergencies, which makes them inappropriate for the summation of a quantity that is required to be finite and positive. For this purpose, it is more appropriate to resort to the method of self-similar approximations [Yukalov(1990a), Yukalov(1990b), Yukalov(1991), Yukalov(1992), Yukalov & Yukalova(1996)] . This is a mathematical method allowing for the construction of effective sums of power-law series, even including divergent series. According to this method, a series (5) is treated as a trajectory of a dynamical system, whose fixed point represents the sought effective sum of the series. In the vicinity of a fixed point, the trajectory becomes self-similar, which gives its name to the method of self-similar approximations.
Here the production parameter b characterizes the type of the influence of the population on the carrying capacity. In the case of creative activity of population, producing additional means for survival, the creation parameter is positive, b>0. And if the population destroys the given carrying capacity, then the destruction parameter is negative (b<0).
By definition, the population is described by a positive variable, so that we will be looking for only positive solutions x(t)>0 for t>0.
This section gives the general conditions for the existence and stability of stationary states of the delayed Eq. (7). The details of such conditions depend on the type of the system characterized by the values of σi. More specific investigations of the evolutionary stable states as well as the overall dynamical regimes will be analyzed in the following sections.
Below we show that varying the parameters of Eq. (7) generates a number of bifurcations and provides a rich variety of qualitatively different solutions. We give a complete classification of all possible solutions demonstrating how the bifurcation control [Chen et al.(2003)] can be realized for this equation.
At the bifurcation point b=1/e, the states coincide: x∗2=x∗3=e. There are no stationary nontrivial states for b>1/e. The existence of the nontrivial states is illustrated in Fig. 1.
Figure 1: Existence of nontrivial stationary states depending on the value of the production parameter b.
which is the basis for the stability analysis. The solution of equation (7) is Lyapunov stable, when the solution to (15) is bounded. A fixed point is asymptotically stable, when the solution to (15) converges to zero, as time tends to infinity. Thus, the conditions of fixed-point stability are prescribed by the convergence to zero of the small deviation described by equation (15).
while for σ1=1 it is always unstable.
According to the existence condition (12), we need to study the stability of the nontrivial states only for coinciding σi.
The following analysis of the stationary states and the solution of the full evolution equation (7) requires to specify the values of σi.
The state x∗3>e is always unstable under σ1=σ2=1.
Thus, there can exist just one stable stationary state x∗2∈(0,e), whose region of stability is given by Eqs. (18) to (20) and shown in Fig. 2.
Figure 2: Stability region (shadowed) for the stationary state x∗2 under σ1=σ2=1.
For these b values, the solution x tends to infinity, as t→∞, for the history x0>x∗3, even for the parameters b and τ in the region of stability of x∗2.
When σ1=σ2=1 and when the production parameter b is non-positive, this means that the population does not produce its carrying capacity but rather destroys it or, in the best case, retains the given capacity value. In this case, an important result for the overall temporal behavior of solutions can be derived rigorously: the population growth has to be limited.
If x0<1, then x→1 from below as t→∞. If x0=1, then x=1 for all t>0. If x0>1, then x→1 from above, as t→∞. So, the solution is always bounded.
implying that x decreases or does not grow.
If the history is such that x0<1, then either x stays always below one, or it grows and reaches one at some moment of time t>0. But x cannot cross the line x=1, since, as is shown above, at the time when x would become ≥1, it has to either stay on this line x=1 or has to decrease. The solution cannot stay forever on the unity line, as far as x=1 is not a stable stationary state, which is x∗2<1 for b<0. This means that there is a moment of time t0<+∞ such that the solution x has to go down for t>t0.
For x0=1, again the solution cannot rise, having a non-positive derivative, and cannot stay forever at this value, which is not a stable fixed point. The sole possibility is that x starts diminishing beyond a finite t0.
When x0>1, then its derivative is non-positive. The solution cannot grow and cannot stay forever at the value that is larger than the fixed point x∗2<1. Hence x must decrease. When diminishing, it reaches the value x=1, where it cannot stay forever, but has to go down beyond a finite time t0.
Thus, for b<0, there always exists such a moment of time, when the solution goes below the value x=1 and can never cross this line from below.
This behavior is illustrated by Fig. 3.
Figure 3: Punctuated unbounded growth, under σ1=σ2=1, for the parameters: b=1>1/e, τ=20, x0=1 (solid line); b=1, τ=15, x0=1 (dashed line); b=0.25, τ=7, x0=10>x∗3=8.613 (dashed-dotted line).
The meaning of such a behavior is clear: for either sufficiently high production parameter or for sufficiently high startup and creative activity, the population (or firm development) can demonstrate unlimited punctuated growth with time.
where the critical time lag is given by (20).
Different regimes of punctuated convergence to the stationary state x∗2 are shown in Fig. 4, which include punctuated growth, punctuated decay, and convergence with plateau reversals. Strongly oscillatory convergence to x∗2 is demonstrated in Fig. 5.
Figure 4: Different types of punctuated convergence to the stationary state x∗2, under σ1=σ2=1, for different parameters: (a) punctuated growth for b=0.25, τ=20, x0=1<x∗2=1.43 (solid line); punctuated decay for b=0.25, τ=20, x0=3>x∗2=1.43 (dashed line); (b) convergence with plateau reversals under negative b=−0.25>−e and τ=20 for x0=2>x∗2=0.816 (solid line) and x0=0.25<x∗2=0.816 (dashed line).
Figure 5: Strongly oscillatory convergence to the stationary state x∗2=0.383, under σ1=σ2=1, with b=−2.5<−e and x0=0.25, for different time lags: (a) τ=20; (b) τ=120.
Figure 6 illustrates this effect of changing the dynamical regime with increasing the time lag above τ∗2 that plays the role of a critical point.
Figure 6: Regime switch from fluctuating convergence to everlasting oscillations, under σ1=σ2=1, with increasing the time lag over its critical value: (a) fluctuating convergence to x∗2=0.35 for b=−3<−e, x0=1, τ=8<τ∗2=8.854; (b) everlasting oscillations for b=−3, x0=1, τ=15>τ∗2=8.854; (c) fluctuating convergence to x∗2=0.175 for b=−10<−e, x0=0.25, τ=1<τ∗2=1.524; (d) everlasting oscillations for b=−10, x0=0.25, τ=20>τ∗2=1.524.
A rather exotic regime develops when the feedback action on the carrying capacity is strongly destructive and the time lag is very long. Then, there appears the regime of everlasting up-down reversals of plateaus located at zero and one, as is shown in Fig. 7.
Figure 7: Everlasting up-down plateau reversals, σ1=σ2=1, for the destruction parameter b=−10≪−e, history x0=0.1, and the time lag τ=100≫τ∗2=1.524.
where x hyperbolically diverges, as t→tc, for some b and x0. The relation between b and x0, for which the divergence occurs, is shown in Fig. 8.
Figure 8: Conditions for the existence of singular solutions under σ1=σ2=−1, depending on the relation between b and x0.
The occurrence of a finite-time singularity is associated with the development of an instability of the system, and the critical time tc corresponds to the time of a change of regime. Concrete interpretations for various dynamical systems with such singularities, corresponding to population growth, mechanical ruptures or fractures, and economic or financial bubbles, have been discussed in detail in many articles [Kapitza(1996), Hern(1999), Korotayev(2007), Yukalov et al.(2004), Yukalov et al.(2009), Yukalov et al.(2012a), Johansen & Sornette(2001), Fogedby & Poutkaradze(2002), Sornette & Andersen(2002), Sornette & Andersen(2006), Andersen & Sornette(2004), Andersen & Sornette(2005)] .
The stability region of x∗3 is presented in Fig. 9. Because the state x∗1 is stable everywhere, hence the stability region of x∗3 corresponds to the region of bistability.
Figure 9: Stability region of the stationary state x∗3 (shadowed), under σ1=σ2=−1. Since x1=0 is always stable, the shadowed region is also the region of bistability.
In the case of loss and cooperation, depending on the parameters b, τ, and the history x0, there can occur the following dynamical regimes: monotonic convergence to a stationary state, convergence with oscillations, everlasting oscillations, and finite-time singularity. All these regimes are demonstrated in Figs. 10 to 13. The summary of all possible solution types is illustrated in the scheme of Fig. 14.
Figure 10: Different types of monotonic convergence to a stationary state, under σ1=σ2=−1, for different parameters: (a) b=1/e, x0=5, τ=0.7>τ∗=0.627 (solid line), τ=0.63>τ∗=0.627 (dashed line), and τ=0.3<τ∗=0.627 (dashed-dotted line); (b) b=0.38, x0=3, τ=0.1 (dashed line) and τ=10 (solid line); (c) b=2, x0=3, τ=10 (solid line) and τ=0.1 (dashed line).
Figure 11: Change of dynamical regimes, under σ1=σ2=−1, for the history x0>x∗3=5.938, with varying time lags: (a) convergence to the stationary state x∗1=0 for b=0.3, τ=20, x0=5.94 (solid line), x0=7 (dashed line), and x0=10 (dashed-dotted line); (b) finite-time singularity at tc=18.09 for b=0.3, x0=7, τ=0.8 in the interval τ1<τ<τ2, with τ1=0.777, τ2=1.398; (c) everlasting oscillations for b=0.3, x0=7, τ=0.68>τ∗3=0.661; (d) transformation of the oscillatory convergence to the state x∗3=5.938 for b=0.3, x0=7 and τ=0.6>τ3=0.31 (solid line) to the monotonic decay to the same state for τ=0.1<τ3=0.31 (dashed line).
Figure 12: Dynamical regimes, under σ1=σ2=−1, for the history x∗2<x0=2<x∗3, with x∗2=1.631, x∗3=5.938, the fixed production parameter b=0.3, varying only the time lag: (a) finite-time singularity at tc=2.42 for τ=10>τ2=0.713277 (dashed line) and τ=0.713278>τ2, with the singularity at tc=3.967 (solid line); (b) everlasting oscillations for τ∗3<τ=0.68<τ2, with τ∗3=0.661, τ2=0.713277; (c) convergence to the stationary state x∗3 for τ1<τ=0.65<τ∗3, with τ1=0.285 (solid line) and τ=0.1<τ1 (dashed line).
Figure 13: Dependence of dynamical regimes, under σ1=σ2=−1, in the case of the destruction parameter b=−0.25, on the history and time lags: (a) finite-time singularity at tc=1.509 for x0=1>x∗2=0.816, τ=10>tc (solid line) and the singularity at tc=1.223 for τ=0.1<tc ( dashed line); (b) monotonic degradation to x∗1=0 for the same b=−0.25, but for x0=0.81<x∗2, τ=5 (solid line), for x0=0.5<x∗2, τ=0.1 (dashed line), and x0=0.81<x∗2, τ=0.1 (dashed-dotted line).
Classification of all possible dynamical regimes, under σ1=σ2=−1, depending on the production parameter b, time lag τ, and history x0.
Contrary to the previous cases, where σ1 and σ2 were equal, now they are of opposite signs, so that σ1σ2=−1.
there is only the stationary state x∗1=0 that is stable for all production parameters b∈(−∞,∞), any time lag τ≥0, and arbitrary history x0≥0. The solutions always monotonically decay to zero, as shown in Fig. 15.
Figure 14: Monotonic decay to zero, under loss and competition (σ1=−1 and σ2=1), with the history x0=5 and time lag τ=10, for b=1 (solid line) and b=−0.1 (dashed line).
When σ1=1,σ2=−1, the solutions always grow with time, exhibiting either unbound increase as t→∞, or a finite-time singularity at a critical time t∗c.
the point of singularity is t∗c=tc(x0,b). But if τ<tc, then the singularity point, defined numerically, is t∗c(x0,b,τ)≥tc, such that t∗c→tc+0, as τ→tc−0. The corresponding finite-time singularities are illustrated in Fig. 16a.
When b>0, there can occur either unbounded growth as t→∞ or a finite-time singularity. If 0≤τ<τ∗, where τ∗=τ∗(x0,b) is defined numerically, the solution unboundedly grows as t→∞. For τ∗≤τ<tc, there exists a critical time t∗c=t∗c(τ)>tc at which the solution diverges. And when τ≥tc, the divergence happens at t∗c=tc, where tc is given by expression (38). The change of the regime for the same production parameter b and history x0, but a varying time lag τ is illustrated in Fig. 16b.
Figure 15: Fig. 16. Behavior of the solutions in logarithmic scale, under σ1=1, σ2=−1, exhibiting either finite-time singularities or unbounded growth at t→∞: (a) finite-time singularity, with the divergence at tc=0.343 for b=−0.1, x0=2, τ=10>tc (solid line) and with the divergence at t∗c=0.256 for b=−0.1, x0=2, τ=0.01<tc (dashed line); (b) the change of regime for b=5, x0=2 and varying time lag, when the finite-time singularity at tc=9.307 for τ=20>tc (dashed line), or at t∗c=9.655>tc for τ∗<τ=9.212<tc, where τ∗=9.2117 (dashed-dotted line), changes to the unbounded growth as t→∞ for τ=9.211<τ∗ (solid line).
We consider Eq. (35) in the sense of Ito, where Wt is the standard Wiener process.
The addition of the noise does not influence much those solutions that do not exhibit finite-time singularities, while the latter can be strongly influenced by even weak noise. Therefore we concentrate our attention on the most interesting case of the noise influence on the solutions with finite-time singularities.
Recall that a finite-time singularity can occur only in the case of cooperation, when σ2=−1. Even rather weak noise can essentially shift the singularity point. Moreover the same noise strength, in different stochastic realizations, shifts the singularity point in a random way. Under the occurrence of a finite-time singularity, the influence of noise turns out to be more important than the variation of the time lag. This is in agreement with the Mao theorem [Mao(1996)] , according to which there can exist a finite range of time lags for which the solution to the differential delay equation is close to that of the related ordinary differential equation. Figure 17 illustrates the influence of noise on the singularity point.
Figure 16: Influence of noise, in the case of b=−0.25 and x0=1, on the point of the finite-time singularity: (a) several realizations of stochastic trajectories, under σ1=σ2=−1 and τ>tc=1.509 for the same noise strength α=0.25; (b) stochastic trajectories, under σ1=1, σ2=−1, with τ≪tc=0.371, for the same noise strength α=0.5; (c) stochastic trajectories, with the parameters as in (b), but for the larger noise strength α=1; (d) singular solutions for the parameters as in (b), but with α=0, for different time lags, τ=0 (solid line), τ=0.01 (dashed line), τ=0.1 (dotted line), and τ=1 (dashed-dotted line).
we find that condition (45) is valid in the following cases.
If b=0, then Eqs. (39) and (40) show that the limit (45) requires the same conditions (48) as for b<0.
The stationary distribution (43) possesses maxima when the effective potential (44) displays minima. The latter correspond to the stable fixed points x∗ of the differential equation for x(t). Combining the analysis of the conditions for the existence of the distribution P(x) with the conditions for the existence of stable fixed points of the delay differential equation (7), we find the following: When the solution x(t), for any history x0 and τ→0, converges to a fixed point, then P(x) exists. Conversely, when there is, at least for some history x0 and τ→0, an unbound solution x(t), diverging either at a finite-time singularity or at increasing time t→∞, then P(x) does not exist. Summarizing, we come to the conclusion.
Statement. The necessary and sufficient condition for the existence of the distribution P(x) is the convergence, for any history x0≥0 and τ→0, of the solution x(t) to a fixed point.
Remark. Note the importance of the limit τ→0. As follows from the analysis of the previous sections, P(x) can exist, though x(t) diverges for some finite τ>0.
Different shapes of the distribution P(x), as a function of x, are shown for the case of a single fixed point x∗1=0 in Fig. 18 and for either the occurrence of the bistability region, with two fixed points x∗1=0 and x∗3>0, or for the case of one nontrivial fixed point x∗2>0 in Fig. 19.
Figure 17: Distribution P(x), as a function of x, in the case of a single fixed point x∗1=0, for the noise strengths α=0.125 (solid line), α=0.25 (dashed line), and α=0.5 (dashed-dotted line), for different parameters: (a) σ1=−1, σ2=1, b=1; (b) σ1=−1, σ2=1, b=−1; (c) σ1=−1, σ2=−1, b=1.
Figure 18: Distribution P(x), as a function of x, in the case of either the bistability region with two fixed points x∗1=0 and x∗3>0 or for one nontrivial fixed point x∗2>0. The parameters are: (a) σ1=σ2=−1, b=0.36, with x∗3=3.397, for α=0.5 (solid line), α=0.75 (dashed line), and α=1 (dashed-dotted line); (b) σ1=σ2=−1, b=0.34, with x∗3=4.268, for the same noise strengths as in (a); (c) σ1=σ2=−1, b=0.33, with x∗3=4.671, for the same noise strengths as in (a); (d) σ1=σ2=1, b=−1, with x∗2=0.567, for the noise strengths α=0.125 (solid line), α=0.25 (dashed line), and α=0.5 (dashed-dotted line).
When the solution diverges at a finite time, this is called a finite-time singularity. As has been discussed above, as well as in a series of papers [Kapitza(1996), Hern(1999), Korotayev(2007), Yukalov et al.(2004), Yukalov et al.(2009), Yukalov et al.(2012a), Johansen & Sornette(2001), Fogedby & Poutkaradze(2002), Sornette & Andersen(2002), Sornette & Andersen(2006), Andersen & Sornette(2004), Andersen & Sornette(2005)] , a finite-time singularity represents a kind of a critical point, where the system experiences a transformation to another dynamic state, similarly to the occurrence of phase transitions in statistical systems [Yukalov & Shumovsky(1990), Sornette(2006)] . Such critical points for complex systems, described by the evolution equations of type (7), depending on applications, can correspond to the points of overpopulation, firm ruin, market crash towards the end of a bubble, earthquakes, and so on. It is evident that the possibility of predicting such disasters would be of great importance. Here, for the case of finite-time singularities, we show that indeed such a prediction can be feasible.
If c1 is negative, this means that, in the vicinity of the initial condition x0, the function x(t) decreases, hence in the near future, we do not expect the occurrence of a singularity, where x(t) would quickly rise. If c1 is positive, then x(t) increases, and the singularity is not excluded. To understand whether it really happens, we need to extrapolate the asymptotic series (49) to longer times. A powerful method for extrapolating asymptotic series has been developed in Refs. [Yukalov et al.(2003), Gluzman et al.(2003), Yukalov & Yukalova(2007a)] , being termed the method of self-similar factor approximants. This method has been proved to be accurate for predicting critical points of different nature, including the critical points for dynamical systems [Yukalov et al.(2003), Gluzman et al.(2003), Yukalov & Yukalova(2007a), Yukalov & Yukalova(2007b), Yukalova et al.(2008)] .
We have investigated the behavior of formula (52) for the different situations studied in the previous sections. We find that, when the evolution equation gives a solution x(t) diverging at a finite time, then the predicted value (52) does approximate the real divergence point tc. When the solution x(t) is bounded, approaching a stationary state, then either A or n is positive, so that the factor approximant (50) does not predict singularities. And, if the solution x(t) tends to infinity for t→∞, then the factor approximant (50) either does not show a finite-time singularity or, in some cases exhibits its appearance. Such artificial singularities can be removed by constructing the factor approximants of higher orders. In order not to complicate the consideration, here we limit ourselves to the second-order factor approximant that does predict the singularity when it really happens for x(t).
In Table 1, we present such error metrics, with fixed x0=1, for varying parameters b.
Figure 19: Predicted singularity times tappc (red line with squares) and real tc (blue line with circles), as functions of the parameter b=−|b| and history x0, under σ1=1, σ2=−1 for: (a) varying |b|, with fixed x0=1; (b) varying x0, with fixed b=−1.
Note that tappc is systematically larger than the true singularity time tc, as can be expected from the fact that the sole information used in the prediction is the quadratic asymptotic representation (49), which necessarily underestimates the full strength of the nonlinear feedback leading to the singularity. Taking third and higher order terms into account would lead to significant improvement of the prediction accuracy. But we believe that using the quadratic asymptotic form (49) is a realistic proxy for the capture of early time dynamics in real life situations. While the relative errors are significant (from 14% to 64% in the investigated cases), we believe that these predictions are useful to provide an approximate estimation of the critical time of the singularity. Taking into account more terms in the asymptotic expansion for x(t) would improve the accuracy of the prediction, however involving more complicated expressions for the critical time.
We have considered the evolution equation describing the population dynamics with functional delayed carrying capacity. The linear delayed carrying capacity, advanced earlier by the authors, has been generalized to the case of a nonlinear delayed carrying capacity. This allowed us to treat 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, when the feedback can be of arbitrary strength. This is contrary to the linear approximation for the capacity, which assumes weak feedback. The nonlinearity essentially changes the behavior of solutions to the evolution equation, as compared to the linear case.
The justification for the exponential form of the nonlinearity is based on the derivation of an effective limit of expansion (5) for the carrying capacity by invoking the self-similar approximation theory.
All admissible dynamical regimes have been analyzed, which happen to 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 has been proved that, for the case of gain and competition, the solutions are always bounded, when the feedback is destructive.
We have studied the influence of additive noise in two cases: (i) on the solutions exhibiting finite-time singularities and (ii) in the presence of stationary solutions. For the former case, we found that even a small noise level profoundly affects the position of the finite-time singularities. For the later case, we have used the Fokker-Planck equation and derived the general condition for the existence of a stationary distribution function.
Finally, we showed that the knowledge of a simple quadratic asymptotic behavior of the early time dynamics of a solution exhibiting a finite-time singularity provides already sufficient information to predict the existence of a critical time, where the solution diverges.
It is necessary to stress that taking into account the nonlinear delayed carrying capacity not merely changes quantitatively the behavior of the solutions to the evolution equation, but also removes artificial finite-time divergence and finite-time death that exist in the equation with the linear form of the carrying capacity. For example, the linear carrying capacity can lead to the appearance of finite-time singularity or finite-time death even in the case of prevailing competition (σ2=1), as is found in [Yukalov et al.(2009), Yukalov et al.(2012a)] . But with the nonlinear carrying capacity, as used in the present paper, these finite-time critical phenomena are excluded. Now, finite-time singularity can occur only in the logically clear case of cooperation (σ2=−1). The reason why the linear approximation for the carrying capacity leads to such artificial singularities and deaths has been explained in Sec. 2.1 of the present paper.
Financial support from the ETH Competence Center ”Coping with Crises in Complex Socio-Economic Systems” is appreciated.
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