Random dynamical systems generated by two Allee maps

TL;DR

This paper investigates the dynamics of systems generated by two Allee maps, highlighting how their behavior varies under deterministic and stochastic conditions, especially noting significant changes with unimodal maps.

Contribution

It introduces analysis of random dynamical systems based on Allee maps, comparing behaviors with and without small random perturbations, and distinguishes effects based on map types.

Findings

Deterministic-like behavior with increasing Allee maps under randomness.

Dramatic behavioral changes in unimodal Allee maps due to stochastic perturbations.

Initial conditions have limited impact on system behavior with unimodal maps.

Abstract

In this paper, we study random dynamical systems generated by two Allee maps. Two models are considered - with and without small random perturbations. It is shown that the behavior of the systems is very similar to the behavior of the deterministic system if we use strictly increasing Allee maps. However, in the case of unimodal Allee maps, the behavior can dramatically change irrespective of the initial conditions.

Figures

Equations

$$x_{n+1}=f(x_n)$$

$$X_{n+1}=\{\begin{array}{ll}f(X_n)& \text{with probability }p,\\ g(X_n)& \text{with probability }1-p\end{array}$$

$$Y_{n+1}=\{\begin{array}{ll}\chi (f(Y_n)+{\epsilon }_n)& \text{with probability }p,\\ \chi (g(Y_n)+{\epsilon }_n)& \text{with probability }1-p,\end{array}$$

$$\chi (x)=\{\begin{array}{ll}0& \text{if }x<0,\\ x& \text{if }x\in [0,b],\\ b& \text{if }x>b.\end{array}$$

$$f(x)=\frac{\rho x^2}{A+x^2},\rho >2\sqrt{A},\rho >0,A>0$$

$$f(Y_n)+{\epsilon }_n<f(w_1)+{\epsilon }_n<f(w_1)+\delta \le f(w_1)+d_1\le f(w_1)+w_1-f(w_1)=w_1.$$

$$f(Y_n)+{\epsilon }_n>f(z_2)+{\epsilon }_n>f(z_2)-\delta \ge f(z_2)-d_2\ge f(z_2)-(f(z_2)-z_2)=z_2$$

$$f(Y_n)+{\epsilon }_n<f(w_3)+{\epsilon }_n<f(w_3)+\delta \le f(w_3)+d_3\le f(w_3)+w_3-f(w_3)=w_3,$$

$$Y_{n+1}=f(Y_n)+{\epsilon }_n>f(Y_n)+\delta -\eta =f(Y_n)+\mu +\eta \ge Y_n+\eta$$

$$f(x)=\frac{Gbx}{(x-T{)}^2+b},b,T>0,G>1,$$

$$P(X_{n+m}\in F|X_n\in E)\ge \lambda$$

$$P(\{n\in N:X_n\in F\}\text{ is unbounded}\mid \{n\in N:X_n\in E\}\text{ is unbounded})=1.$$

$$P(X_{n+1}\in [0,M_f]\mid X_n\in [0,M])\ge p>0.$$

$$P(X_{n+1}\in [0,A_f)\mid X_n\in [0,C))\ge 1-p>0.$$

$$P(X_{n+m}<A_f\mid X_n\in (K_f-\eta ,K_f+\eta ))>(p(1-p){)}^m>0.$$

$$P(X_{n+r}\in (K_f-\eta ,K_f+\eta )\mid X_n\in [C,M_f])>p^r>0.$$

$$P(X_{n+m}<D\mid X_n\in (K_f-\eta ,K_f+\eta ))>(p(1-p){)}^m>0$$

$$P(X_{n+2}\in [0,A_f-\xi ]\mid X_n\in [0,A_f+\xi ))\ge p(1-p)>0.$$

$$P(X_{n+2}\in [A_f+\xi ,M_f]\mid X_n\in (A_f-\xi ,A_f+\xi ))\ge p(1-p)>0,$$

$$P(Y_{n+1}\in [0,M_f]\mid Y_n\in [0,M])\ge pP({\epsilon }_n<0)>0,$$

$$P(Y_{n+1}\in I_1\mid Y_n\in I_2)\ge p(1-p)P({\epsilon }_n\in (-\mu ,\mu ))>0.$$

$$P(Y_{n+2}<A_f\mid Y_n\in I_2)\ge (p(1-p){)}^{2}P({\epsilon }_n\in (-\mu ,\mu ))P({\epsilon }_{n+1}\in (-\nu ,\nu ))>0.$$

Full text

1 Introduction

The Allee effect, which was first described by W.C. Allee [1932], is a biological phenomenon characterized by a correlation between the population size and the per capita population growth rate. In the literature, the most frequently mentioned reason of this effect is the difficulty of finding mates in a smaller population (Boukal & Berec, [2002]). There are several possible scenarios for the behavior of the population size according to the strength of the Allee effect (see Boukal & Berec, [2002]).

The Allee effect was investigated from various points of view using models of dynamical and semi-dynamical systems. Interesting results were obtained for non-autonomous periodic systems. We mention several recent achievements. A special attention was given to the study of the Beverton-Holt equation. In connection with a periodic model based on this equation, two conjectures were formulated in Cushing & Henson, [2002]. These conjectures were proved in the later results of Elaydi and Sacker [2005] and also for more general systems in Kon, [2004] (see also Elaydi & Sacker, [2006]). It was shown that periodic fluctuations in the environment have, in a certain sense, a deleterious effect on the average population in the corresponding models. Periodic fluctuations in a context of economical models with an Allee effect were studied in Luís et al., [2009]. A class of unimodal Allee maps was introduced in Luís et al., [2010] where properties and stability of the three fixed points in non-autonomous periodic dynamical systems with period 2 were studied.

Some recent works take into account also randomly varying systems with the Allee effect. The stochastic stability of such systems was investigated in Haskell & Sacker, [2005] and Bezandry et al., [2008]. Environmental stochasticity in a connection with Allee effect was examined and persistence, asymptotic extinction and conditional persistence for stochastic difference equations was analyzed in Roth & Schreiber, [2014].

In this paper, only the extinction-survival scenario is studied. If the population size drops below a certain critical value, the population becomes extinct in the long run. If the size exceeds the critical value, the population is generally able to survive. Therefore the function (called an Allee map) $f:[0,b]\to[0,b]\ (b\leq\infty)$ describing population dynamics with this effect

[TABLE]

($x_{n}$ is the population size in the $n$-th generation) must satisfy the following conditions: {itemlist}[(3)]

$f$ has three fixed points - zero, which is a stable fixed point, an unstable threshold point $A_{f}>0$ and a fixed point $K_{f}>A_{f}$, which can (but does not have to) be stable,

$f(x)<x$ for all $x\in(0,A_{f})\cup(K_{f},b]$,

$f(x)>x$ for all $x\in(A_{f},K_{f})$.

We consider a pair of continuous Allee maps $f,g$ and two random dynamical systems generated by these maps. The first system is given by

[TABLE]

while the other one,

[TABLE]

contains also perturbations. In (3), $\{\varepsilon_{n}\}_{n=0}^{\infty}$ are continuous i.i.d. random variables with a positive density on the interval $(-\delta,\delta)$, $\delta>0$ and $\chi:R\to R$ is a function defined as follows:

[TABLE]

More formally (see Bhattacharya & Majumdar, [2007]), we can write $X_{n+1}=H_{n}(X_{n})$, where $\{H_{n}\}_{n=1}^{\infty}$ are i.i.d. random functions such that $P(H_{n}=f)=p$ and $P(H_{n}=g)=1-p$ for some $p\in(0,1)$ and $X_{0}=x_{0}\in[0,b]$, similarly for model (3). Given $X_{n}$, the variable $X_{n+1}$ does not depend on $X_{n-1},X_{n-2},\ldots$, and thus the process $\{X_{n}\}_{n=0}^{\infty}$ is a Markov process with $X_{0}=x_{0}\in[0,b]$. The same is true for the process $\{Y_{n}\}_{n=0}^{\infty}$ with $Y_{0}=y_{0}\in[0,b]$.

Braverman [2011] considered a model similar to (3), i.e., with one function and scarce random perturbations. It was shown that such perturbations can, under some conditions, cause the extinction or survival of the population regardless of its initial size.

Section 2 will focus on the behavior of models (2) and (3) for strictly increasing functions $f$ and $g$. Unimodal functions $f$ and $g$ will be analyzed in Section 3 - in both models, the population becomes extinct under some conditions, irrespective of its initial size. We can thus observe a behavior similar to the Parrondo’s paradox Harmer & Abbott, [1999].

2 Strictly increasing Allee maps

In this section, we assume that the functions $f$ and $g$ are continuous and strictly increasing. An example of such Allee map,

[TABLE]

can be found in Hoppensteadt, [1982]. It is also a special case of a new model presented in Elaydi & Sacker, [2009].

The behavior of the non-stochastic model (i.e., model (1)) is very simple. If $x_{0}>A_{f}$, then $\lim\limits_{n\to\infty}x_{n}=K_{f}$ and if $x_{0}<A_{f}$, then $\lim\limits_{n\to\infty}x_{n}=0$. Before we formulate theorems about models (2) and (3), it is useful to mention the following lemma and its corollary.

Lemma 2.1**.**

Let $\{X_{n}\}_{n=0}^{\infty}$ be a Markov process and let $X_{n}\in B$ for any $n\in N$. Assume that $B_{1}\subset B$ is such that $P(X_{n+k}\in B_{1}\mid X_{n}\in B)\geq\lambda$ for some $\lambda>0$ and $k\in N$. Then $P(\exists n_{0}\in N:X_{n_{0}}\in B_{1})=1$.

A proof of this lemma can be found in Janková & Smítal, [1995].

Corollary 2.2**.**

Let $I_{1}\subset I_{2}$ be intervals such that the following conditions are satisfied: {arabiclist}[(2)]

$f(I_{i})\subset I_{i}$* and $g(I_{i})\subset I_{i}$, $i=1,2$.*

*For any $x\in I_{2}$ $f^{m}(x)\in I_{1}$ or $g^{m}(x)\in I_{1}$ for some $m\in N$.

Then $P(\exists n_{0}\in N:X_{n}\in I_{1}\ \forall n\geq n_{0}\mid x_{0}\in I_{2})=1$.*

Theorem 2.3**.**

Let $A_{f}<A_{g}<K_{f}<K_{g}$. Then {romanlist}[(iii)]

if $x_{0}\leq A_{f}$, then $p_{0}\equiv P(\lim_{n\to\infty}X_{n}=0)=1$,

if $x_{0}\geq A_{g}$, then $p_{1}\equiv P(\exists n_{0}\in N:X_{n}\in[K_{f},K_{g}]\ \forall n\geq n_{0})=1$,

if $x_{0}\in(A_{f},A_{g})$, then $p_{0}>0$, $p_{1}>0$ and $p_{0}+p_{1}=1$.

Proof 2.4**.**

{romanlist}

[(iii)]

Since $\lim_{n\to\infty}g^{n}(A_{f})=0$, for any $\eta>0$ there exists an $m\in N$ such that $g^{m}(A_{f})<\eta$. However, the function $g$ is strictly increasing, so $g^{m}$ is also strictly increasing, hence for any $x\leq A_{f}$ we have $g^{m}(x)<\eta$. It follows that we can apply Corollary 1 to intervals $I_{1}=[0,\eta)$ and $I_{2}=[0,A_{f}]$. This can be done for an arbitrary small $\eta>0$, hence if $x_{0}\leq A_{f}$, then $p_{0}=1$.

*If $x_{0}\in[A_{g},K_{g}]$, then $X_{n}\in[A_{g},K_{g}]$ for all $n\in N$. Let us denote $B_{1}$ the set $[K_{f},K_{g}]$. If $x_{0}\in B_{1}$ then $K_{f}=f(K_{f})\leq f(x)\leq x\leq K_{g}$ and also $K_{f}\leq x\leq g(x)\leq g(K_{g})=K_{g}$. Hence $f(B_{1})\subset B_{1}$ and $g(B_{1})\subset B_{1}(*)$.

Next, $A_{g}<f(A_{g})<K_{f}$, so $g^{n}(f(A_{g}))\nearrow K_{g}$ and hence for some $m\in N$ $g^{m}(f(K_{g}))\in[K_{f},K_{g}]$. Moreover, function $g^{m}\circ f$ is increasing, so it follows that $g^{m}(f(x))\in[K_{f},K_{g}]$ for all $x\in[A_{g},K_{g}]$.

Combining $(*)$ and Lemma 1, applied to the sets $B=[A_{g},K_{g}]$ and $B_{1}$ and $\lambda=p(1-p)^{m}$, we get $p_{1}=1$.

If $x_{0}\geq K_{g}$, then $f^{n}(x_{0})\searrow K_{f}$ and we can apply Corollary 1 to the intervals $I_{1}=[K_{f},K_{g}]$ and $I_{2}=[K_{f},x_{0}]$.*

*If $x_{0}\in(A_{f},A_{g})$, then $f^{m}(x_{0})>A_{f}$ and $g^{m}(x_{0})<A_{f}$ for some $m\in N$. Hence $p_{0}>0$ and $p_{1}>0$.

It remains to show that $P(X_{n}\in(A_{f},A_{g})\ \forall n\in N)=0$. Let $C=\frac{A_{f}+A_{g}}{2}$. Again, for some $m\in N$ we have $f^{m}(C)>A_{g}$ and since $f$ is increasing, $f^{m}(x)>A_{g}$ for any $x\geq C$. Similarly $g^{m}(x)<A_{f}$ for all $x\leq C$. Hence we can apply Lemma 1 to the sets $[0,b]$ and $[0,A_{f}]\cup[A_{g},b]$ with $\lambda=p^{m}(1-p)^{m}$.*

Theorem 2.5**.**

Let $A_{f}<A_{g}<K_{f}<K_{g}$. Take a $\delta>0$ such that the sets {itemlist}[(3)]

$U_{1}=\{x\in(0,A_{f}):\min(x-f(x),x-g(x))\geq\delta\},$**

$U_{2}=\{x\in(A_{g},K_{f}):\min(f(x)-x,g(x)-x)\geq\delta\},$**

$U_{3}=\{x\in(K_{g},b):\min(x-f(x),x-g(x))\geq\delta\}$*

are nonempty. Assume that for every $x\in[0,b]$ there exists an $x^{*}\geq x$ such that $\min(x^{*}-f(x^{*}),x^{*}-g(x^{*}))\geq\delta$. Then {romanlist}[(iii)]*

if $y_{0}\in[0,z_{1}]$, then $p_{0}\equiv P(\exists n_{0}\in N:Y_{n}\in[0,w_{1})\ \forall n\geq n_{0})=1$,

if $y_{0}\in[w_{2},b]$, then $p_{1}\equiv P(\exists n_{0}\in N:Y_{n}\in(z_{2},w_{3})\ \forall n\geq n_{0})=1$,

*if $y_{0}\in(z_{1},w_{2})$, then $p_{0}>0$, $p_{1}>0$ and $p_{0}+p_{1}=1$,

where $w_{i}=\inf U_{i},\ i=1,2,3$ and $z_{j}=\sup U_{j},\ j=1,2$ (see Fig. 2).*

Proof 2.6**.**

{romanlist}

[(iii)]

Let $d_{1}=\min(w_{1}-f(w_{1}),w_{1}-g(w_{1}))$. If $Y_{n}\in[0,w_{1})$, then we have

[TABLE]

Similar inequalities hold also for the function $g$, hence $Y_{n+1}\in[0,w_{1})$. Similarly, if $Y_{n}\in[0,z_{1}]$, then $Y_{n+1}\in[0,z_{1}]$. Moreover, $f^{n}(z_{1})\searrow 0$, therefore we can apply Lemma 1 to the sets $B=[0,z_{1}]$ and $B_{1}=[0,w_{1})$ with $\lambda=(pP(\varepsilon_{n}<0))^{k}$ for some $k\in N$.

Let $d_{2}=\min(f(z_{2})-z_{2},g(z_{2})-z_{2})$ and $d_{3}=\min(w_{3}-f(w_{3}),w_{3}-g(w_{3}))$. If $Y_{n}\in(z_{2},w_{3})$, then $Y_{n+1}\in(z_{2},w_{3})$, because

[TABLE]

and

[TABLE]

similarly for the function $g$. If $Y_{n}>w_{3}$, then by the assumption there exists a $z_{3}>Y_{n}$ such that $\min(z_{3}-f(z_{3}),z_{3}-g(z_{3}))\geq\delta$. Again, it can be shown that if $Y_{n}\in(z_{2},z_{3})$, then $Y_{n+1}\in(z_{2},z_{3})$, and so we can apply Lemma 1 (like in the case (i)) to the sets $B=(z_{2},z_{3})$ and $B_{1}=(z_{2},w_{3})$, because $f^{n}(z_{3})\searrow K_{f}\in(z_{2},w_{3})$. The case $Y_{n}>w_{2}$ can be shown similarly.

*If $z_{1}<Y_{n}\leq A_{f}$, then the probability that $Y_{n+k}\leq z_{1}$ for some $k\in N$ is obviously non-zero. However, we will show in the following that also $Y_{n+m}>A_{f}$ with non-zero probability for some $m\in N$.

Let us denote $h$ the function defined by $h(x)=\min(x-f(x),x-g(x))$. Since $f$ and $g$ are continuous functions, $h$ is also continuous and it follows that it attains its maximum on the interval $[Y_{n},A_{f}]$ (denote this maximum $\mu$). Next, let $\eta=\frac{\delta-\mu}{2}$. Since $z_{1}=\sup U_{1}$ and $\operatorname{argmax}\limits_{x\in[Y_{n},A_{f}]}h(x)>z_{1}$, it follows that $\mu<\delta$, and so $0<\eta<\delta$. Hence the probability that $\varepsilon_{n}$ takes a value from $(\delta-\eta,\delta)$ is non-zero (because the density of $\varepsilon_{n}$ is positive on $(-\delta,\delta)$).

Without loss of generality, assume that $Y_{n}-f(Y_{n})\leq\mu$, and hence $Y_{n}\leq f(Y_{n})+\mu$ (if it is not true for the function $f$, then it is true for $g$). Hence with non-zero probability we have*

[TABLE]

*(the second equality is obtained by the definition of $\eta$). Similarly, we can continue with $Y_{n+1},Y_{n+2}$, etc. and with the same $\eta$. Finally, if an $m\in N$ is sufficiently large, $Y_{n+m}>Y_{n}+m\eta>A_{f}$ with a positive probability. But if $Y_{n+m}>A_{f}$, then obviously $Y_{n+m+l}>w_{2}$ with a non-zero probability for some $l\in N$.

Similarly, it can be shown that if $A_{g}\leq Y_{n}<w_{2}$, then with a non-zero probability $Y_{n+k}>w_{2}$ for some $k\in N$ and also $Y_{n+m}<z_{1}$ for some $m\in N$. The case $Y_{n}\in(A_{f},A_{g})$ and the fact that $P(Y_{n}\in(z_{1},w_{2})\ \forall n\in N)=0$ can be shown as in Theorem 2.3.*

For $A_{f}<A_{g}<K_{g}<K_{f}$ or for the cases where $A_{f}=A_{g}$ or $K_{f}=K_{g}$, very similar statements hold.

3 Unimodal Allee maps

In this section, we assume that there exists a $B_{f}\in(A_{f},K_{f})$ such that the function $f$ is strictly increasing on $(0,B_{f})$ and strictly decreasing on $(B_{f},b)$ (denote $M_{f}=f(B_{f})$ the maximum of the function $f$). We also assume that the function $f$ is continuous. An example of such Allee map is the function

[TABLE]

see Asmussen, [1979].

Similarly, let $g$ be a continuous unimodal function with the maximum $M_{g}$ attained at the point $B_{g}$. Let $M=\max(M_{f},M_{g})$. For every $x\in[0,b]$ it holds that $\max(f(x),g(x))\leq M$. Thus in this section we assume that $b=M$. Again, if $x_{0}<A_{f}$ in the deterministic model (1), then $\lim_{n\to\infty}x_{n}=0$, but if $x_{0}>A_{f}$, the behavior of the model can vary according to the function $f$. However, if $x_{0}>A_{f}$, a sufficient condition for the survival of the population is $f(M_{f})>A_{f}$ (in this case the population size $x_{n}$ never drops below the threshold value $A_{f}$).

As in Section 2, we first need the following corollary of Lemma 2.1.

Corollary 3.1**.**

Let the sets $E$ and $F$ be such that there exists $m\in N$ with the property

[TABLE]

for some $\lambda>0$. Then

[TABLE]

(If the process returns infinitely many times to the set $E$, then it almost surely returns infinitely many times to the set $F$).

Theorem 3.2**.**

Let $A_{f}<A_{g}$, $f$ be differentiable and $|f^{{}^{\prime}}(x)|<1$ for every $x\in(B_{f},M_{f})$. Assume that there exist functions $h_{1},h_{2},\ldots h_{m}\in\{f,g\}$ for which $h_{1}\circ h_{2}\circ\ldots\circ h_{m}(K_{f})<A_{f}.$ Then $P(\lim_{n\to\infty}X_{n}=0)=1$ for every $x_{0}$.

Example. Consider two functions $f(x)=\frac{2.2x}{(x-3)^{2}+2}$ and $g(x)=\frac{1.3x}{(x-3.3)^{2}+1}$ (see Fig 4).

Here $A_{f}=3-\sqrt{0.2}\approx 2.553$, $K_{f}=3+\sqrt{0.2}\approx 3.447$, $A_{g}=3.3-\sqrt{0.3}\approx 2.752$ and $K_{g}=3.3+\sqrt{0.3}\approx 3.848$ (hence $A_{f}<A_{g}$). Next, $g(f(g(K_{f})))\approx 1.876<A_{f}$, so in this case $h_{1}=g,h_{2}=f$ and $h_{3}=g$. Moreover, it can be shown that the function $f$ is concave on the interval $(B_{f},M_{f})$, hence its first derivative is decreasing on this interval. Since $f^{{}^{\prime}}(B_{f})=0$ and $f^{{}^{\prime}}(M_{f})\approx-0.475>-1$, the condition $|f^{{}^{\prime}}(x)|<1$ is satisfied for every $x\in(B_{f},M_{f})$. Therefore, for the process $\{X_{n}\}_{n=0}^{\infty}$ generated by these two functions, we have $P(\lim_{n\to\infty}X_{n}=0)=1$ for every $x_{0}\in[0,b]$.

Proof 3.3**.**

Since $M_{f}$ is the maximum of the function $f$, we obviously have

[TABLE]

Next, $g(A_{f})<A_{f}$ (because $A_{f}<A_{g}$) and since $g$ is continuous, there is some $C\in(A_{f},B_{f})$ such that $g(x)<A_{f}$ for every $x\in[A_{f},C)$. Hence

[TABLE]

Denote $h(x)$ the function defined by $h(x)=h_{1}\circ h_{2}\circ\ldots\circ h_{m}(x)$. This function is continuous (because $f$ and $g$ are continuous), hence there is an $\eta>0$ such that $h(x)<A_{f}$ for any $x\in(K_{f}-\eta,K_{f}+\eta)$. Consequently

[TABLE]

Moreover, $\lim_{n\to\infty}f^{n}(x)=K_{f}$ for any $x\in(A_{f},M_{f}]$ (since $|f^{{}^{\prime}}(x)|<1$ for every $x\in(B_{f},M_{f})$), and hence there is an $r\in N$ such that $f^{r}(x)\in(K_{f}-\eta,K_{f}+\eta)$ for any $x\in[C,M_{f}]$, and so

[TABLE]

Now we can gradually apply Corollary 3.1 to the sets {itemlist}[(4)]

$[0,M]$* and $[0,M_{f}]$ (by (12)),*

$[0,M_{f}]$* and $[0,A_{f})\cup[C,M_{f}]$ (if $X_{n}\in[0,C)$, we can use (13) and if $X_{n}\in[C,M_{f}]$, then $f(X_{n})\in[C,M_{f}]$),*

$[0,A_{f})\cup[C,M_{f}]$* and $[0,A_{f})\cup(K_{f}-\eta,K_{f}+\eta)$ (by (15) and by the fact that $f([0,A_{f}))\subseteq[0,A_{f})$),*

$[0,A_{f})\cup(K_{f}-\eta,K_{f}+\eta)$* and $[0,A_{f})$ (by (14)).

This means that the process $\{X_{n}\}_{n=0}^{\infty}$ returns to $[0,A_{f})$ infinitely many times, and hence, from previous results (in Section 2), we have $P(\lim_{n\to\infty}X_{n}=0)=1$.*

Theorem 3.4**.**

Let $A_{g}<A_{f}$, $f$ be differentiable and $|f^{{}^{\prime}}(x)|<1$ for every $x\in(B_{f},M_{f})$. Assume there exist functions $h_{1},h_{2},\ldots,h_{m}\in\{f,g\}$ for which $h_{1}\circ h_{2}\circ\ldots\circ h_{m}(K_{f})<A_{f}$. If, in addition, $f(g(A_{f}))\neq A_{f}$, then for every $x_{0}$ $P(\lim_{n\to\infty}X_{n}=0)=1$.

Proof 3.5**.**

Since $h(K_{f})=h_{1}\circ\ldots\circ h_{m}(K_{f})<A_{f}$, it follows that $h(K_{f})<D$ for some $D<A_{f}$. As $h$ is continuous, we have

[TABLE]

*for some $\eta>0$. Since $f(g(A_{f}))\neq A_{f}$, two cases are possible: either $f(g(A_{f}))<A_{f}$ or $f(g(A_{f}))>A_{f}$.

If $f(g(A_{f}))<A_{f}$, then $f(g(A_{f}))<C$ for some $C<A_{f}$ and from the continuity of $f\circ g$, we have $f(g(x))<C$ for any $x\in(A_{f}-\xi,A_{f}+\xi)$, where $0<\xi<\min(A_{f}-C,A_{f}-D)$.*

Consequently

[TABLE]

Again, we can apply Corollary 3.1 to the sets {itemlist}[(5)]

$[0,M]$* and $[0,M_{f}]$ (by (12)),*

$[0,M_{f}]$* and $[0,A_{f}-\xi]\cup[A_{f}+\xi,M_{f}]$ (by (17) if $X_{n}\in[0,A_{f})$ and by the fact, that $f([A_{f}+\xi,M_{f}])\subseteq[A_{f}+\xi,M_{f}]$ if $X_{n}\in[A_{f}+\xi,M_{f}]$),*

$[0,A_{f}-\xi]\cup[A_{f}+\xi,M_{f}]$* and $[0,A_{f}-\xi]\cup(K_{f}-\eta,K_{f}+\eta)$ (as in the preceding theorem),*

$[0,A_{f}-\xi]\cup(K_{f}-\eta,K_{f}+\eta)$* and $[0,A_{f}-\xi]$ (by (16), because $D<A_{f}-\xi$),*

$[0,A_{f}-\xi]$* and $[0,A_{g})$ (since $f^{n}(A_{f}-\xi)\searrow 0$, and so for some $s\in N$ $f^{s}(x)<A_{g}$ for an arbitrary $x\in[0,A_{f}-\xi]$).

It follows that the process $\{X_{n}\}_{n=0}^{\infty}$ returns to the set $[0,A_{g})$ infinitely many times, hence $P(\lim_{n\to\infty}X_{n}=0)=1$.

If $f(g(A_{f}))>A_{f}$, then $f(g(A_{f}))>C$ for some $C>A_{f}$ and also $f(g(A_{f}))\leq M_{f}$ (since $M_{f}$ is the maximum of the function $f$). Therefore*

[TABLE]

where $0<\xi<\min(A_{f}-D,C-A_{f})$. The rest of the proof is similar to the proof of the first case.

Example. Let $f(x)=\frac{1.155x}{(x-2.8)^{2}+1.05}$ and $g(x)=\frac{1.3x}{(x-2.9)^{2}+1}$ (see Fig. 6).

Here $A_{f}\approx 2.476,$ $K_{f}\approx 3.124,$ $A_{g}\approx 2.352$ and $K_{g}\approx 3.448$ (hence $A_{g}<A_{f}$). Further, $f(g(A_{f}))\approx 2.986\neq A_{f}$ and $f(g(K_{f}))\approx 2.041<A_{f}$. Moreover, the function $f$ is again concave on $(B_{f},M_{f})$ and $f^{{}^{\prime}}(M_{f})\approx-0.989>-1$, hence $|f^{{}^{\prime}}(x)|<1$ for every $x\in(B_{f},M_{f})$. Thus, from Theorem 3.4 it follows that $P(\lim_{n\to\infty}X_{n}=0)=1$.

Since $f^{n}(x)\in[A_{f},M_{f}]$ for every $n\in N$ and for an arbitrary $x_{0}\in(A_{f},M_{f})$, the population persists in the deterministic model (model (1)) generated by the function $f$; similarly the population persists in the model generated by $g$. However, if we combine these models, the population becomes almost surely extinct. This resembles results known as the Parrondo’s paradox (see e.g. Harmer & Abbott, [1999]).

A very similar theorem holds for the process $\{Y_{n}\}_{n=0}^{\infty}$.

Theorem 3.6**.**

Let $f$ be a differentiable function and $|f^{{}^{\prime}}(x)|<1$ for an arbitrary $x\in(B_{f},M_{f})$. Assume there exist functions $h_{1},h_{2},\ldots,h_{m}\in\{f,g\}$ for which $h_{1}\circ h_{2}\circ\ldots\circ h_{m}(K_{f})<A_{f}$. If, in addition, the set

$U=\{x\in[0,\min(A_{f},A_{g})]:\min(x-f(x),x-g(x))\geq\delta\}$**

is nonempty (see Fig. 7), then $P(\exists n_{0}\in N:Y_{n}\in[0,\inf U)\ \forall n\geq n_{0})=1$ for every $x_{0}$.

The proof is almost the same as the proof of Theorem 3.2 and Theorem 3.4 - inequality (12) can be modified to

[TABLE]

similarly for inequality (13). Inequality (14) can be shown as follows:

Suppose that $h_{1}(h_{2}(K_{f}))<A_{f}$ for some $h_{1},h_{2}\in\{f,g\}$ (the proof is the same for $m$ functions). Since $h_{1}$ is continuous, we have $h_{1}(x)<A_{f}$ for an arbitrary $x\in I_{1}=(h_{2}(K_{f})-\eta_{1},h_{2}(K_{f})+\eta_{1})$ for some $\eta_{1}>0$. Next, from the continuity of $h_{2}$, there exists an $\eta_{2}>0$ such that $h_{2}(x)\in I_{1}$ for every $x\in I_{2}=(K_{f}-\eta_{2},K_{f}+\eta_{2})$. Now let $Y_{n}\in I_{2}$. Then $h_{2}(Y_{n})\in I_{1}$ and since $I_{1}$ is an open set, there exists a $\mu\in(0,\delta)$ such that $(h_{2}(Y_{n})-\mu,h_{2}(Y_{n})+\mu)\subset I_{1}$. It follows that

[TABLE]

Next, if $Y_{n+1}\in I_{1}$, then $h_{1}(Y_{n+1})<A_{f}$, and so there exists a $\nu\in(0,\delta)$ such that $h_{1}(Y_{n+1})+\nu<A_{f}$. Hence we obtain

[TABLE]

Inequalities (15), (16), (17) and (18) can be shown similarly - from the continuity of the functions $f$ and $g$ and by the fact that the density of $\varepsilon_{n}$ is positive on $(-\delta,\delta)$.

Remark. The condition $f(g(A_{f}))\neq A_{f}$ in Theorem 3.4 is not necessary in this case - Corollary 3.1 can be applied directly to the sets $[0,M_{f}]$ and $[0,A_{f}-\frac{\delta}{2}]\cup[A_{f}+\frac{\delta}{2},M_{f}]$.

4 Concluding remarks

{arabiclist}

[(5)]

In Section 2, we showed that models (2) and (3) lead to similar results as the deterministic model, if we work with two strictly increasing Allee maps (recall that we distinguish only between extinction and survival here). If the initial population size is greater than some threshold, the population persists; if it is smaller than another threshold, the population becomes extinct. The only difference from the non-stochastic case is the region between these two thresholds - if the initial population size is in this region, both scenarios (extinction or survival) are possible.

In Theorems 2.3 and  2.5 only the inequality $A_{f}<A_{g}<K_{f}<K_{g}$ was considered. If the order of these values is different, proofs and results are very similar. The only difference is in the case $A_{f}<K_{f}<A_{g}<K_{g}$ (or $A_{g}<K_{g}<A_{f}<K_{f}$). For these two orderings of the values, it can be easily shown that the population becomes almost surely extinct.

If $\delta$ does not satisfy the conditions of Theorem 2.5 or Theorem 3.6 (i.e. $\varepsilon$ can take relatively high values), then it is not possible to predict the long-term behavior of the system. Its values can increase even from zero to high numbers and vice versa.

In Section 3, we showed that under some conditions the population becomes almost surely extinct. This is caused by the fact that the population size almost surely drops below the critical value $\min(A_{f},A_{g})$, which leads to the extinction. Therefore, it would be interesting to calculate the expected value of the time after which the system drops below this critical value. Generally, this can be a very difficult problem, because this time depends on the given functions $f$ and $g$ and also on the probability $p$ with which we choose functions $f$ and $g$. However, if we fix the functions and the probability, the expected time can be estimated by simulations. We examined the expected time $T(p)$ in the case where the functions were the same as in the second example for different probabilities and for $x_{0}\in(A_{f},M_{f})$. As previously mentioned (in the second example), if we consider only one function ($f$ or $g$), the population size never drops below the critical value, hence $T(p)\to\infty$ for $p\to 0$ and $p\to 1$. For $p\in(0,1)$ the estimates are shown in Fig. 8. A question is whether it is possible to calculate these values exactly and in more general situations.

In Section 3, we considered only specific conditions for unimodal Allee maps, which have interesting consequences. It would be interesting to study other situations.

\nonumsection

Acknowledgments Supported by the VEGA Grant No. 2/0047/15 from the Slovak Scientific Grant Agency (Kováč, Janková) and the UK/344/2016 grant from Comenius University in Bratislava (Kováč).