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We study a population model with strong and weak Allee effect driven by internal noise and external noise. Firstly, a single-species population model with Allee effect under environmental colored noise is established, then stable and unstable states are analyzed and interpreted in biology. After that, stationary probability distribution (SPD) of population is derived based on Fokker-Planck equation. Next, mean first-passage time (MFPT) is defined in order to quantify the transition between extinction state and survival state with Allee effect. It is found that population will not extinct when weak Allee effect exists. It is not beneficial to survival of the population with the increase of Allee threshold no matter whether strong Allee effect or weak Allee effect. When strong Allee effect occurs, the correlation time of multiplicative noise plays a positive role in survival of population, while the correlation time of additive noise has a negative effect. Crucially, the phenomenon of resonant activation is firstly discovered in population dynamics with Allee effect. The conclusions we obtain can be applied to the further research of population dynamics in ecology.
The impact of noise on population model has been widely studied over several decades due to its crucial theoretical and experimental significance. The analysis usually based on bistable model driven by multiplicative noise and additive noise. The selection of noise is directly related to the accuracy of research. Accordingly, the choice of noise seems particularly critical for the study. Normally, Gaussian white noise is chosen as the stochastic perturbation of model. However, it is more reasonable to use colored noise as stochastic disturbances in the study.
Much effort devoted to the impacts of colored noise on the biological model. The influence of environmental colored noise on the single-species population system was investigated by Spanio. They concentrated on the phase transitions and how the existence of time-correlations noise affects these phase transitions in Ref. [1]. The effects of colored noise on the tumor model were discussed by Bose,[2] Xu[3] and Fiasconaro[4] in modern medicine. In their work, the impacts of both noise intensity and stability index on tumor system were explored and then colored noise can enhance stability of system was found. Recently, the properties of delayed bistable model with cross-correlated colored noise and mean first-passage time of bistable dynamics models driven by colored noises were investigated by Jin et al.[5–6] Studies explored the properties of system and demonstrated that noise intensity and coupling strength could all affect mean first-passage time. Stationary probability distribution and mean first-passage time can well present the properties of model.[7–9] Additional, a three species ecosystem consisting of a prey, a predator and a top predator was studied by Das.[10] The barrier crossing dynamics with non-Gaussian noises was analyzed and resonant activation was observed by Goswami.[11] Closely, a stochastic model driven by colored noise was discussed by Zhang,[12] the researches of Fokker-Planck function and mean first-passage time were carried out. Thus the research of stationary probability distribution and mean first passage time is a crucial step toward studying of the population model.
Allee effect is a phenomenon that can not be ignored in population growth. Sometimes, population growth can not achieve exponential growth when initial size is really small (
Allee effect has attracted extensive attention due to its important biological significance in recent years. The influence of Allee effect on predator-prey system at discrete times was studied by Celik and Duman.[20] The change of equilibrium points move from unstable states to stable states under Allee effect is explored in their study; The extinction conditions of isolated population with Allee effect were discussed by Mendez et al.,[21] which provided valuable theoretical help in biology and medicine; The major conclusion what Allee effect could enhance population stability was demonstrated by Scheuring[22] through numerical simulations of host microorganisms. Additional, Allee effect also applied in the study of biotic invasion. The species invasion of stochastic population model with Allee effect was investigated by Ackleh et al.[23] They formulated the relationships between initial population size, migration rates, Allee threshold and the possibility of species invasion, then they explored the influence of Allee effect on species invasion. The impact of Allee effect on exotic species establishment is also concerned by Petrovskii.[24] It concluded that Allee effect could increase the system spatiotemporal complexity through the research. Recently, how the noise affects the behaviors of Truscott-Brindley system under weak Allee effect was analysed.[25] Studies investigated and discussed the phenomenon of stochastic excitement and Canard explosion. A new result of traveling wave solutions for a biological invasion model involving density dependent migration and Allee effect was reported by Sun.[26] Closely, a prey-predator model with strong Allee effect in the prey growth function was considered by Sen,[27] and they conducted a extensive study of the overall dynamics of the system and all possible global bifurcations that the system could undergo were explored. The single-species model and predator prey model with Allee effect was driven by colored noise terms are investigated by Sun.[28]
In this work, we focus on the impact of colored noise on single-species population model under strong Allee effect and weak Allee effect. The goal of our work is to study the model by discussing in detail how colored noise affects stationary probability distribution and mean first-passage time under different Allee effects (strong and weak).
The other parts of contents are summarized as follows. In Sec.
The deterministic system model is formally introduced and then random noise terms are added to the model in this section. For simplicity, the dimensionless formulations are adopted in the paper. Now, the most common model for describing the Allee effect of a single species population is given
In contrast when
The potential functions when Allee threshold m takes different values in strong or weak Allee effect are plotted in Figs.
The population system is disturbed by external environment. The weather conditions, temperature, amount of food, number of natural enemies and growth rate will affect the population system. A slight change may even cause major changes in the population system. Therefore, we use noise to represent stochastic perturbations that the population may suffer.
Next, the random perturbations are introduced to the model. In the system, multiplicative noise is identified as internal perturbations, whereas additive noise is equivalent to external perturbations. External perturbations are supposed to originated from the environment. In contrast, internal perturbations are thought to be generated within the system. For instance, the stochastic perturbations of population size and intrinsic growth rate are regarded as internal noise. Now consider a population model driven by multiplicative noise and additive noise, which follows the Langevin equation:
To investigate the influence of noise on stationary probability distribution, the expression for stationary probability distribution is calculated based on Fokker-Planck equation (FPE). According to the Novikov theorem[34] and Foxʼs approach, approximate Fokker-Planck equation can be written:[33]
Here, it should be noted that approximate Fokker-Planck equation is valid under the condition
It can also be represented as the following form:
After calculation, it is obtained
Firstly, for the purpose of check the validity of approximation method used in the derivation, we use Milstein method to take numerical simulation of SPD. The analytical results and simulation results of SPD are shown in Fig.
We consider the effects of noises intensities on the SPD under strong Allee effect in Figs.
Figure
Figure
Figure
Figure
In order to study the model in more depth, it is essential to estimate the amount of time between shifts from one stable state to another. It contributes to quantify the influence of noise on the state transitions between stable states. This time is called the first passage time. When the first passage time is averaged in many realizations, the result is called mean first-passage time. The longer mean first-passage time explains that the state is more steady.[36]
In this paper, we focus on the transition of population from the state of survival to the state of extinction. Hence, the mean first-passage time from one stable state xs2 = K to another xs1 = 0 under strong Allee effect is given by the following expression:[37–38]
Figure
Comparing with Fig.
Figure
Figure
We discuss the impacts of colored noise on the population model with Allee effect in this paper. We firstly establish the single-species population model. Next, we explore the stationary probability distribution and mean first-passage time of system. Meantime, we also discuss in detail how colored noise affects stationary probability distribution and mean first-passage time.
It is found that noise intensity, correlation time, Allee threshold and intrinsic growth rate can all affect stationary probability distribution and mean first-passage time. It is noteworthy that effect of many parameters on the population is often restricted by other parameters, which restrict each other and jointly affect the survival and stability of the population. Whether it happens strong Allee effect or weak Allee effect, the smaller Allee threshold is better for the survival and stability of population. Importantly, the occurrence of weak Allee effect has positive influence on the survival of population. These conclusions play a significant role in the study of ecological problems in the future. In general, the increase in intensity and correlation time of multiplicative noise all favors the stability of population under strong Allee effect. In contrast when weak Allee effect occurs, the increase in correlation time of multiplicative noise will have benefits for the survival of population, whereas the increase in intensity of multiplicative noise may lead to the extinction of population at this time. Surprisingly, the phenomenon of resonant activation is firstly discovered in population dynamics with Allee effect. This discovery makes the research more novel and our conclusions become more abundant.
Allee effect can deeply affect the integrity of population or dynamic status of the whole community. Some scholars pointed out dynamic instability like Allee effect can be used as the impetus of evolution for highly developed species in the system. Obviously, Allee effect plays a pivotal role in biological evolution and our research of Allee effect is particularly necessary.
In brief, noise and Allee effect are two principle elements in ecology. In this work, phase transition is induced by noise. Besides, the phenomenon of resonant activation is caused by both noise and Allee effect. Previously, it is demonstrated that noise can enhance the stability of prototype dynamical system and resonant activation.[40] While in the population system, the phenomenon of resonant activation is the result of interaction between noise and Allee effect.
Further work on analysing impact of colored noise on population is needed. In practice, colored noises are correlated, they interact with each other and affect the system together. Hence, the coupling strength of noise is another important factor affecting the model. The correlated colored noise can be considered in the population model as the next step. Additionally, internal noise can affect directly on the intrinsic growth rate, which is also one aspect of our next work can be improved. Our work is to investigate the population dynamics based on time variation. Besides, there are many researches of population dynamics based on spatial changes. The influence of time delay and spatial diffusion on herbivore system were studied by Li[41] and Sun.[42] The stability behaviors of a marine prey-predator model was considered by Gazi,[43] they studied the reaction diffusion equation and analyzed the influence of diffusion on the stability. A predator prey model with spatial motion was studied by Sun,[44] they found isolation degree of spatial patterns have a significant effect on the persistence of population. For this, our future work can be extended to the analysis of the dynamic behaviors of population system based on time variation and spatial motion. More specific investigates of Allee effect and colored noise on population are major challenges that worth extensive attention and intensive research. Still, our work is a crucial step toward resolving these challenges.
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