Modelling and bifurcation analysis in a hybrid bioeconomic system with gestation delay and additive Allee effect
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Abstract
In this paper, we investigate a delayed differential algebraic prey–predator system, where commercial harvesting on predator and additive Allee effect on prey are considered. A discrete time delay is utilized to represent gestation delay of the predator population. Positivity of solutions and uniform persistence of system are discussed. In the absence of time delay, by taking economic interest as a bifurcation parameter, some sufficient conditions associated with additive Allee effect and economic interest are derived to show that the proposed system undergoes singularityinduced bifurcation around the interior equilibrium. In the presence of time delay, combined dynamic effects of time delay and additive Allee effect on population dynamics are discussed in the case of positive economic interest of commercial harvesting. Existence of Hopf bifurcation and local stability switch around the interior equilibrium are studied as gestation delay crosses the critical value. Furthermore, properties of Hopf bifurcation are investigated based on the center manifold theorem and the norm form of a delayed singular system. Existence of global continuation of periodic solutions bifurcating from interior equilibrium is discussed by using a global Hopf bifurcation theorem. Numerical simulations are provided to show consistency with theoretical analysis.
Keywords
Gestation delay Economic interest Singularityinduced bifurcation Additive Allee effect Hopf bifurcation1 Introduction
In the 1930s, W.C. Allee proposed the concept of Allee effect from experimental studies and extensively investigated ecological significance of animal aggregations [1]. It is well known that Allee effect is highly relevant to reduction in mating success, reduced inbreeding efficiency, suppressed social thermoregulation [2, 3], and some other biological or ecological reasons which can be found in [4, 5] and the references therein. It may cause a positive feedback between a component of individual fitness and either number or density of conspecifics. From ecological perspective, Allee effect can be categorized as strong Allee effect and weak Allee effect, respectively. If some population undergoes the strong Allee effect, then the population must surpass certain threshold to sustainable survival. However, the population threshold does not exist for the population with weak Allee effect.
We will extend the work in [12] by incorporating commercial harvesting on predator into system (1). \(E(t)\) represents the commercial harvesting effort on predator at time t, w represents the harvesting reward coefficients, c represents the cost per unit harvesting effort for unit weight of predator. v is the economic interest of commercial harvesting on predator. Based on system (1) TR and TC in Eq. (2), it is easy to show that \(\mathrm{TR}=w E(t)P(t)\) and \(\mathrm{TC}=cE(t)\).
Remark 1.1
Recently, it has been shown that commercial harvesting on a prey–predator system with Allee effect has a strong impact on population dynamics (see [7, 10, 15, 16, 17, 18, 19] and the references therein). However, dynamical behavior due to variation of economic interest of commercial harvesting is not discussed in [7, 10, 15, 16, 17, 18]. Only dynamic effects of strong Allee effect on population dynamics are considered in [19], the weak Allee effect case is not investigated in [19]. On the other hand, the reproduction of predator after predating prey is not instantaneous but will be mediated by some time lag required for gestation of the predator population. Hence, it is necessary to investigate combined dynamics of time delay and additive Allee effect on population dynamics of a harvested prey–predator system with commercial harvesting. Although the harvested prey–predator system with Allee effect has attracted a great deal of attention, to authors’ best knowledge, little work has been done on combined dynamic effects of time delay and additive Allee effect on population dynamics of the harvested prey–predator system with commercial harvesting.
Remark 1.2
Since the algebraic equation in (3) includes no differentiated variables, the third row in matrix \(\Xi(t)=\left [ {\scriptsize\begin{matrix}{}1& 0 & 0 \cr 0& 1 &0\cr 0&0&0 \end{matrix}} \right ] \) has a corresponding zero row.
The remaining sections of this paper are organized as follows. Positivity of solutions and uniform persistence of system (3) are investigated in the second section. In the third section, in the absence of time delay, the existence of singularityinduced bifurcation is investigated under the case of additive Allee effect on prey. In the absence of time delay, combined dynamic effects of time delay and additive Allee effect on population dynamics are discussed, local stability switch around interior equilibrium and the existence of Hopf bifurcation are also discussed. In the fourth section, properties of Hopf bifurcation are investigated. Existence of global continuation of periodic solutions bifurcating from interior equilibrium is discussed by using a global Hopf bifurcation theorem. In the fifth section, numerical simulations are provided to support theoretical findings. Finally, this paper ends with a conclusion.
Remark 1.3
The dynamical model proposed in [12] is composed of ordinary differential equations, and it is utilized to study interaction mechanism of a prey–predator system with additive Allee effect. Compared with the system established in [12], an algebraic equation is introduced into system (3), which concentrates on dynamic effect of economic interest of commercial harvesting on population dynamics and provides a straightforward way to investigate complex dynamics due to variation of economic interest. Furthermore, a discrete time delay, which represents gestation delay of the predator population, is incorporated into system (3). Consequently, compared with the work done in [12], we can investigate combined dynamic effects of time delay and additive Allee effect on population dynamics by analyzing the local stability and bifurcation phenomenon of system (3) in this paper.
2 Positivity and uniform persistence
In this section, positivity of solutions and uniform persistence of system (3) with initial conditions (4) will be studied.
Proof
For solutions of system (3), it is easy to show that \(F_{i}:R^{3+1}_{+}\rightarrow R^{3}\) is locally Lipschitz and satisfies the condition \(F_{i}>0\), where \(F_{i}\) (\(i=1,2,3\)) can be found in (5). Due to the lemma in [20] and Theorem A.4 in [21], all solutions of system (3) with initial conditions (4) exist uniquely, and each component of solution remains within the interval \([0, U_{0})\) for some \(U_{0}>0\). Standard and simple arguments show that any solution of system (3) always exists and stays positive. □
Lemma 2.2
([22])
 (i)
if\(a_{1}>a_{2}\), then\(\lim_{t\rightarrow\infty}u(t)=\frac {a_{1}a_{2}}{a_{3}}\),
 (ii)
if\(a_{1}< a_{2}\), then\(\lim_{t\rightarrow\infty}u(t)=0\).
Theorem 2.3
Ifτis bounded, \(e^{\frac{b\tau}{k_{1}}}< k_{1}\)and\(we^{2(\tau+k_{2}\tau)}>c\)hold, then system (3) with initial conditions (4) is uniformly persistent.
Proof
Based on the second equation of system (3), if follows from Theorem 2.1 and (6) that \(\dot{P}(t)\leq k_{2}P(t)\).
If \(w e^{2(\tau+k_{2}\tau)}>c\) and τ are bounded, then it is easy to show that \(M_{3}>0\) is bounded, and then there exists \(T_{7}>T_{6}\) such that \(E(t)\geq M_{3}\) holds for \(t>T_{7}\).
3 Local stability analysis
Furthermore, we will investigate the existence of positive roots of Eq. (13).
Lemma 3.1
Proof
By simple computations, it is easy to show that the maximum and minimum values of function \(h(x)\) are \(L_{1}=\eta_{2}+2\sqrt{ \vert \eta_{1} \vert ^{3}}\) and \(l_{1}=\eta_{2}2\sqrt{ \vert \eta_{1} \vert ^{3}}\), respectively.
When \(\eta_{3}<0\) and \(\eta_{1}\geq0\), by virtue of \(\dot {h}(x)=3(x^{2}+\eta_{1})\geq0\), we have \(h(x)\) is strictly increasing and continuous in \([0,+\infty)\), which yields \(h(x)\geq h(0)=\eta _{2}\). Consequently, \(h(x)\) has a positive root.
When \(\eta_{3}<0\) and \(\eta_{1}\leq0\), it is easy to show that Eq. (13) has a positive root.
When \(\eta_{2}>0\), it is easy to derive that \(\eta_{1}<0\), otherwise \(h(x)=x^{3}+3\eta_{1}x+\eta_{2}\) may not be equal to zero.
If \(\eta_{2}^{2}+4\eta^{3}_{1}=0\), then Eq. (13) has a positive root of multiplicity two. Furthermore, if \(\eta_{2}^{2}+4\eta^{3}_{1}<0\), then Eq. (13) has two positive roots.
When \(\eta_{2}=0\), we have that \(\eta_{1}<0\), otherwise \(h(x)=x^{3}+3\eta_{1}x+\eta_{2}\) may not be equal to zero. Hence, Eq. (13) has a unique positive root. □
Based on the above analysis, sufficient conditions associated with the existence of interior equilibria of system (3) with strong Allee effect and weak Allee effect can be concluded in Lemma 3.2 and Lemma 3.3, respectively.
Lemma 3.2
 (i)If either of the following inequalities holds:then system (3) has a unique interior equilibrium$$ \textstyle\begin{cases} 2k_{1}+1ab>0,\quad 0< a< m< m^{*}_{2}, \\ 2k_{1}+1ab< 0,\quad 0< \max\{a,m^{*}_{2}\}< m, \end{cases} $$(14)$$ S_{s}^{*}= \bigl(N_{s}^{*},P_{s}^{*},E_{s}^{*} \bigr)= \biggl( x_{1}\xi_{1}, \frac{c}{w}, k_{2} \biggl(1\frac{c}{w(x_{1}\xi_{1}+k_{3})} \biggr) \biggr) . $$
 (ii)If either of the following inequalities holds:then the following conclusions associated with the existence of interior equilibria of system (3) hold:$$ \textstyle\begin{cases} 2k_{1}+1ab>0, \quad 0< \max\{a,m^{*}_{2}\}< m< m^{*}_{1}, \\ 2k_{1}+1ab< 0,\quad 0< b< m< \min\{m^{*}_{1},m^{*}_{2}\}, \end{cases} $$(15)
 system (3) has two interior equilibria:$$\begin{aligned}& S_{s1}^{*}= \bigl(N_{s1}^{*},P_{s1}^{*},E_{s1}^{*} \bigr)= \biggl( x_{1}\xi_{1}, \frac{c}{w}, k_{2} \biggl(1\frac{c}{w(x_{1}\xi_{1}+k_{3})} \biggr) \biggr) , \\& S_{s2}^{*}=(N_{s2}^{*},P_{s2}^{*},E_{s2}^{*})= \biggl( x_{2}\xi_{1}, \frac{c}{w}, k_{2}\biggl(1\frac{c}{w(x_{2}\xi_{1}+k_{3})}\biggr) \biggr) ; \end{aligned}$$
 system (3) has a unique interior equilibriumwith\(\eta_{2}^{2}+4\eta_{1}^{3}=0\).$$ {S}_{s3}^{*}= \bigl({N}_{s3}^{*},{P}_{s3}^{*},{E}_{s3}^{*} \bigr)= \biggl( \sqrt{\eta_{1}}, \frac{c}{w}, k_{2} \biggl(1\frac{c}{w(\sqrt{\eta_{1}}+k_{3})} \biggr) \biggr) $$

 (iii)If\(a< m< m_{1}^{*}\), system (3) has a unique interior equilibrium$$ {S}_{s}^{*}= \bigl({N}_{s}^{*},{P}_{s}^{*},{E}_{s}^{*} \bigr)= \biggl( \sqrt{3\eta_{1}},\frac{c}{w},k_{2} \biggl(1\frac{c}{w(\sqrt{3\eta_{1}+k_{3}})} \biggr) \biggr) . $$
Lemma 3.3
 (i)If either of the following inequalities holds:then system (3) has a unique interior equilibrium$$ \textstyle\begin{cases} 2k_{1}+1ab>0,\quad 0< m< \min\{a,m_{2}^{*}\}, \\ 2k_{1}+1ab< 0,\quad 0< m_{2}^{*}< m< a, \end{cases} $$(16)$$ S_{w}^{*}= \bigl(N_{w}^{*},P_{w}^{*},E_{w}^{*} \bigr)= \biggl( x_{1}\xi_{1},\frac{c}{w},k_{2} \biggl(1\frac{c}{w(x_{1}\xi_{1}+k_{3})} \biggr) \biggr) . $$
 (ii)If either of the following inequalities holds:then the following conclusions associated with the existence of interior equilibria of system (3) hold:$$ \textstyle\begin{cases} 2k_{1}+1ab>0,\quad 0< m_{2}^{*}< m< \min\{a,m^{*}_{1}\}, \\ 2k_{1}+1ab< 0,\quad 0< m< \max\{a,m^{*}_{1},m_{2}^{*}\}, \end{cases} $$(17)
 system (3) has two interior equilibria as follows:$$\begin{aligned}& S_{w1}^{*}= \bigl(N_{w1}^{*},P_{w1}^{*},E_{w1}^{*} \bigr)= \biggl( x_{1}\xi_{1},\frac{c}{w},k_{2} \biggl(1\frac{c}{w(x_{1}\xi_{1}+k_{3})} \biggr) \biggr) , \\& S_{w2}^{*}= \bigl(N_{w2}^{*},P_{w2}^{*},E_{w2}^{*} \bigr)= \biggl( x_{2}\xi_{1},\frac{c}{w},k_{2} \biggl(1\frac{c}{w(x_{2}\xi_{1}+k_{3})} \biggr) \biggr) ; \end{aligned}$$
 system (3) has a unique interior equilibriumwhere\(\eta_{2}^{2}+4\eta_{1}^{3}=0\).$$ S_{w3}^{*}= \bigl(N_{w3}^{*},P_{w3}^{*},E_{w3}^{*} \bigr)= \biggl( \sqrt{\eta_{1}},\frac{c}{w},k_{2} \biggl(1\frac{c}{w(\sqrt{\eta_{1}+k_{3}})} \biggr) \biggr) , $$

 (iii)If\(0< m<\max\{a,m^{*}_{1}\}\), then system (3) has a unique interior equilibrium$$ S_{w}^{*}= \bigl(N_{w}^{*},P_{w}^{*},E_{w}^{*} \bigr)= \biggl( \sqrt{3\eta_{1}},\frac{c}{w},k_{2} \biggl(1\frac{c}{w(\sqrt{3\eta_{1}+k_{3}})} \biggr) \biggr) . $$
According to the above analysis, the existence conditions for interior equilibrium in the case of strong Allee effect are concluded in Lemma 3.4.
Lemma 3.4
According to the above analysis, the existence conditions for interior equilibrium in the case of weak Allee effect are concluded in Lemma 3.5.
Lemma 3.5
3.1 Case I: system (3) with strong Allee effect
By taking v as a bifurcation parameter, the existence of singularityinduced bifurcation and local stability switch around \(S^{*}_{s}\) and \(\tilde{S}^{*}_{s}\) will be investigated due to variation of v in Theorem 3.6.
Theorem 3.6
When\(\tau=0\)and\(m>a\), system (22) undergoes singularityinduced bifurcation around\(S^{*}_{s}\), \(v=0\)is a bifurcation value. Whenvincreases though 0, system (22) is unstable around\(S^{*}_{s}\)and\(\tilde{S}^{*}_{s}\)in the case of zero and positive economic interest, respectively.
Proof
By defining \(g_{3}(X_{1}(t),X_{2}(t),v)=D_{X_{2}}g_{2}(X_{1}(t),X_{2}(t),v)=w P(t)c\).
Based on three items (23), (24), and (25) computed above, the existence theorem of singularityinduced bifurcation (Theorem in [25]) holds, hence system (22) undergoes singularityinduced bifurcation around \(S^{*}_{s}\) and the bifurcation value is \(v=0\).
By virtue of (7), (8), and (19) and simple computations, it can be obtained that \(\frac{{G}_{1}}{{G}_{2}}>0\), \(\frac{\tilde{G}_{1}}{\tilde {G}_{2}}>0\). Based on Theorem 3 in [25], when v increases through 0, one eigenvalue of system (22) moves from \(\mathbb{C}^{}\) to \(\mathbb{C}^{+}\) along the real axis by diverging through ∞. Hence, when v increases through 0, system (22) is unstable around \(S^{*}_{s}\) and \(\tilde{S}^{*}_{s}\) in the case of zero and positive economic interest, respectively. □
Theorem 3.7
Proof
If \(2k_{1}< a<\min\{m, \frac{k_{1}m}{k_{1}1}\}\), \(0< v< w k_{3}(k_{2}+1+\sqrt{k_{2}+1})\) hold, then system (3) has at least an interior equilibrium \(\tilde{S}^{*}_{s}\). Furthermore, if \(a>k_{3}\), \(k_{1}>k_{3}\), \(c\leq1\), and \(0< v<(1c)k_{2}+ck_{2}k_{2}w w \tilde {P}^{*}_{s}\) hold, then it is obtained that \(n_{2s}^{2}n_{1s}^{2}<0\), which guarantees that Eq. (27) has a pair of purely imaginary roots of the form \(\pm i\sigma^{*2}_{s}\).
By using Butler’s lemma [26], system (3) is locally asymptotically stable around \(\tilde{P}^{*}_{s}\) when \(0<\tau<\tau_{1c}^{*}\).
If \(0< v<\min\{k_{2}w \tilde{P}^{*}_{s}ck_{2}, (1c)k_{2}+ck_{2}k_{2}w w \tilde{P}^{*}_{s}\}\), then it follows from simple computations that \(\Theta>0\). Consequently, if \(0< v< v_{1}\) holds, where \(v_{1}\) is defined in Theorem 3.7, then the transversality condition holds and system (3) undergoes Hopf bifurcation around \(\tilde{S}^{*}_{s}\) when \(\tau=\tau_{1c}^{*}\). □
3.2 Case II: system (3) with weak Allee effect
When \(\tau=0\) and \(0< m< a\), by taking v as a bifurcation parameter, the existence of singularityinduced bifurcation and local stability switch around \(S^{*}_{w}\) and \(\tilde{S}^{*}_{w}\) will be investigated due to variation of v in Theorem 3.8.
Theorem 3.8
When\(\tau=0\)and\(0< m< a\), system (22) undergoes singularityinduced bifurcation around\(S^{*}_{w}\), \(v=0\)is a bifurcation value. Whenvincreases though 0, system (22) is unstable around\(S^{*}_{w}\)and\(\tilde{S}^{*}_{w}\)in the case of zero and positive economic interest, respectively.
Theorem 3.9
4 Properties of Hopf bifurcation
In this section, τ is regarded as a bifurcation parameter. Properties of Hopf bifurcation around interior equilibrium \(\tilde {S}^{*}_{w}\) in the case of weak Allee effect are discussed. By using the similar analysis, symmetric analysis about the properties of Hopf bifurcation around interior equilibrium \(\tilde{S}^{*}_{s}\) in the case of strong Allee effect can be also obtained, which is omitted in this section.
If φ is a given function in \(C([{\tau}, 0],\mathbb{R}^{2})\) and \(Y(\varphi)\) is the unique solution of linearized equation \(\dot {Y}(t)=L_{\upsilon}(Y_{t})\) of Eq. (29) with initial function φ at zero, then the solution operator \(\tilde{T}(t): \mathbb {C}\rightarrow\mathbb{C}\) is defined as \(\tilde{T}(t)\varphi =Y_{t}(\varphi)\), \(t\geq0\).
Suppose that \(r(\vartheta)=(1,\beta)^{T}e^{i\sigma_{w}^{*} \tau _{1d}^{*} \vartheta}\) is an eigenvector of \(B(0)\) corresponding to \(i\sigma_{w}^{*} \tau_{1d}^{*}\), which derives \(B(0)r(0)=i\sigma _{w}^{*} \tau_{1d}^{*} r(\vartheta)\). By virtue of \(B(0)\), (30), (31), and (32), it follows from \(r(1)=r(0)e^{i\sigma_{w}^{*}\tau _{1d}^{*}}\) that \(\beta=\frac{i\sigma_{w}^{*} (b_{11}+b_{13})e^{i\sigma_{w}^{*} \tau_{1d}^{*}}}{b_{12}}\).
Similarly, it follows from simple computations that the eigenvector of \(B^{*}\) corresponds to \(i\sigma_{w}^{*}\tau_{1d}^{*} \vartheta\), which gives that \(\beta^{*}=\frac{i\sigma_{w}^{*}}{b_{22}}\).
Theorem 4.1
 (i)
If\(\gamma_{2}>0\) (\(\gamma_{2}<0\)), then Hopf bifurcation is supercritical (subcritical), and the bifurcating periodic solutions exist for\(\tau>\tau_{1d}^{*}\) (\(\tau>\tau_{1d}^{*}\));
 (ii)
Bifurcating periodic solutions are stable (unstable) if\(\iota_{2}<0\) (\(\iota_{2}>0\));
 (iii)
Period increases (decreases) if\(T>0\) (\(T<0\)).
We define \((\bar{Z}, \tau, \mu)\) as a center provided that \((\bar{Z}, \tau, \mu)\in\mathcal{N}\) and \(\Delta(\bar{Z}, \tau, \mu)=0\). The center \(\Delta(\bar{Z}, \tau, \mu)\) is relevant to be isolated when it is the only center in some neighborhood of it. The global Hopf bifurcation theorem for general functional delayed differential equations introduced in [28] is stated as follows.
Lemma 4.2
 (i)
\(\mathcal{L}_{(\bar{Z}, \tau, \mu)}\)is unbounded,
 (ii)
\(\mathcal{L}_{(\bar{Z}, \tau, \mu)}\), \(\mathcal{L}_{(\bar {Z}, \tau, \mu)}\cap\Gamma\)is finite and\(\sum_{(\bar{Z}, \tau , \mu)\in\mathcal{L}(\bar{Z}, \tau, \mu)\cap\mathcal{N}}\gamma _{m}(\bar{Z}, \tau, \mu)=0\)for\(m=1,2,\ldots\) , \(\gamma_{m}(\bar{Z}, \tau, \mu)\)is themth crossing number of\((\bar{Z}, \tau, \mu)\).
It is easy to show that if (ii) of Lemma 4.2 is not true, then \(\mathcal {L}\) is unbounded. Hence, if the projections of \(\mathcal{L}_{(\bar{Z}, \tau, \mu)}\) onto zspace and onto μspace are bounded, then the projection of \(\mathcal{L}_{(\bar{Z}, \tau, \mu)}\) is unbounded. Furthermore, if we can show that the projection of \(\mathcal{L}_{(\bar {Z}, \tau, \mu)}\) onto τspace is away from zero, then the projection of τspace must include \([\tau, \infty)\).
Theorem 4.3
If\(2ak^{3}_{3}< m< a\), \(\max\{2k_{1}, m\}< a<1\), \(k^{3}_{3}< k_{1}\), \(0< v< v_{2}\)hold (\(v_{2}\)has been defined in Theorem 3.9), then for\(\tau_{1}>\tau_{1d}^{*}\)and\(\sigma_{w}^{*}\)defined in Theorem 3.9, system (29) has at least one periodic solution.
Proof
It is easy to show that \((\tilde{S}^{*}_{w}, \tau, \frac {2\pi}{\sigma_{w}^{*}})\) is an isolated center of system (29). Let \(\mathcal{L}_{(\tilde{S}^{*}_{w}, \tau, \frac{2\pi}{\sigma _{w}^{*}})}\) denote a connected component passing through \((\tilde {S}^{*}_{w}, \tau, \frac{2\pi}{\sigma_{w}^{*}})\) in Γ. It follows from Theorem 3.9 of this paper that \(\mathcal{L}_{(\tilde {S}^{*}_{w}, \tau, \frac{2\pi}{\sigma_{w}^{*}})}\) is nonempty.
Let \(\Delta(\tilde{S}^{*}_{w}, \tau, \rho)(\lambda)\) represent the characteristic matrix of system (29) around \(\tilde{S}^{*}_{w}\). Based on the discussion in Sect. 3.2 of this paper, it can be verified that \((\tilde{S}^{*}_{w}, \tau, \frac{2\pi}{\sigma_{w}^{*}})\) is an isolated center, and there exist \(\epsilon>0\), \(\delta_{1}>0\), a smooth curve \(\lambda: (\tau\delta_{1}, \tau+\delta _{1})\rightarrow\mathbb{C}\) such that \(\vert \Delta(\tilde {s}^{*}_{w}, \tau, \mu)(\lambda) \vert =0\), \(\vert \lambda (\tau)i\sigma_{w}^{*} \vert <\epsilon\) for all \(\tau\in[\tau _{1d}^{*}\delta_{1}, \tau_{1d}^{*}+\delta_{1}]\).
It is easy to show that \(\vert \Delta(\tilde{S}^{*}_{w}, \tau _{1}, \mu)(\eta+\frac{2\pi i}{\mu}) \vert =0\) if and only if \(\eta=0\), \(\tau_{1}=\tau_{1d}^{*}\), \(\mu=\frac{2\pi}{\sigma_{w}^{*}}\).
It follows from Lemma 4.2 of this paper that the connected component \(\mathcal{L}_{(\tilde{S}^{*}, \tau_{1d}^{*}, \frac{2\pi}{\sigma _{w}^{*}})}\) passing through \((\tilde{S}^{*}, \tau_{1d}^{*}, \frac {2\pi}{\sigma_{w}^{*}})\) in Γ is unbounded.
Next, we will show that the projection \(\mathcal{L}_{(\tilde{S}^{*}, \tau_{1d}^{*}, \frac{2\pi}{\sigma_{w}^{*}})}\) onto τspace is \([\bar{\tau}, \infty)\), where \(\bar{\tau}<\tau_{1d}^{*}\). If \(2ak^{3}_{3}< m< a\), \(\max\{2k_{1}, m\}< a<1\), \(k^{3}_{3}< k_{1}\), \(0< v< v_{2}\), then system (29) without time delay has no nontrivial periodic solution. Hence, the projection of \(\mathcal {L}_{(\tilde{S}^{*}, \tau_{1d}^{*}, \frac{2\pi}{\sigma_{w}^{*}})}\) onto τspace is away from 0.
It is assumed that the projection of \(\mathcal{L}_{(\tilde{S}^{*}, \tau _{1d}^{*}, \frac{2\pi}{\sigma_{w}^{*}})}\) onto τspace is bounded, which implies the projection of \(\mathcal{L}_{(\tilde{S}^{*}, \tau_{1d}^{*}, \frac{2\pi}{\sigma_{w}^{*}})}\) onto τspace is included in a bounded interval \((0, \tau_{1d}^{*})\). By using \(\frac {2\pi}{\sigma_{w}^{*}}<\tau_{1d}^{*}\), we have \(\mu<\tau_{1d}^{*}\) for \((Z, \tau, \mu)\) belonging to \(\mathcal{L}_{(\tilde{S}^{*}, \tau _{1d}^{*}, \frac{2\pi}{\sigma_{w}^{*}})}\), which implies that the projection of connected component \(\mathcal{L}_{(\tilde{S}^{*}, \tau _{1d}^{*}, \frac{2\pi}{\sigma_{w}^{*}})}\) onto μspace is bounded. Hence, it leads to a contradiction, which means that the projection of \(\mathcal{L}_{(\tilde{S}^{*}, \tau_{1d}^{*}, \frac{2\pi }{\sigma_{w}^{*}})}\) onto τspace is \([\tau_{1d}^{*}, \infty )\), where \(\bar{\tau}\leq\tau_{1d}^{*}\). □
5 Numerical simulation
Numerical simulations are carried out to show combined dynamic effects of time delay and additive Allee effect on population dynamics. Parameters are partially taken from the numerical simulations in [12].
Numerical simulation I: strong Allee effect. Parameters are taken as follows: \(a=0.25\), \(b=0.1\), \(k_{1}=0.035\), \(k_{2}=0.1\), \(k_{3}=0.2\), \(w=4\), \(c=1\), and \(m=0.3\) with appropriate units. By simple computations, it can be obtained that there exists a unique interior equilibrium provided that \(0< v<0.0264\). In the following numerical simulation, \(v=0.01\) is arbitrarily selected within \((0, 0.0264)\) which is enough to merit theoretical analysis in this paper. By using given parameters and simple computations, the unique interior equilibrium of system (3) with strong Allee effect is as follows: \(\tilde{S}_{s}^{*}=(0.6029,0.2891,0.0640)\).
Numerical simulation II: weak Allee effect. Parameters are taken as follows: \(a=0.4\), \(b=0.5\), \(k_{1}=0.3\), \(k_{2}=0.125\), \(k_{3}=0.2\), \(w=5\), \(c=1\), and \(m=0.24\) with appropriate units. By simple computations, it can be obtained that there exists a unique interior equilibrium provided that \(0< v<0.0194\). In the following numerical simulation, \(v=0.01\) is arbitrarily selected within \((0, 0.0194)\) which is enough to merit theoretical analysis in this paper. By using given parameters and simple computations, the unique interior equilibrium of system (3) with weak Allee effect is as follows: \(\tilde{S}_{w}^{*}=(0.2643,0.4226,0.0089)\).
6 Conclusion
In this paper, a delayed singular biological system with additive Allee effect and commercial harvesting is established, which extends the work done in [12] by incorporating gestation delay for the predator population. Positivity of solutions and uniform persistence of the proposed system are discussed in Theorem 2.1 and Theorem 2.3, respectively. Some sufficient conditions for the existence of interior equilibrium in the case of strong and weak Allee effect are investigated in Lemma 3.2 to Lemma 3.5. In the absence of time delay, on account of variation of economic interest of commercial harvesting, we reveal that system (3) undergoes singularityinduced bifurcation around interior equilibrium, and the proposed system is unstable when economic interest increases through zero, which can be found in Theorem 3.6 and Theorem 3.8 in the case of strong Allee effect and weak Allee effect, respectively. In the presence of time delay, by analyzing the associated characteristic equation of system (3), we reveal that when gestation delay crosses the corresponding critical value, the interior equilibrium of the system loses local stability, and system (3) undergoes Hopf bifurcation, which can be found in Theorem 3.7 and Theorem 3.9 in the case of strong Allee effect and weak Allee effect, respectively. By using the center manifold theorem and the norm form of a delayed singular system, we investigate the properties of Hopf bifurcation in Theorem 4.1. The global continuation of Hopf bifurcation is investigated in Theorem 4.3. Finally, numerical simulations are provided to validate theoretical analysis obtained in this paper.
It follows from the analytical findings in Theorem 3.6 and Theorem 3.8 that a singularityinduced bifurcation occurs and local stability switches when economic interest increases through zero, and it can be practically interpreted that the population density dramatically increases beyond environment capacity during a short time period. It follows from the analytical findings in Theorem 3.7 and Theorem 3.9 that local stability will switch when time delay crosses critical value. It can be practically interpreted that commercially harvested population density shows periodic fluctuation and may even arrive at a very low population density during some time, which is disadvantageous to sustainable survival of each population and sustainable exploitation of certain economic population.
The dynamical model proposed in [12], composed of ordinary differential equations, is utilized to study the interaction mechanism of a prey–predator system with additive Allee effect. Compared with the system established in [12], an algebraic equation is introduced into system (3), which concentrates on dynamic effect of economic interest of commercial harvesting on population dynamics and provides a straightforward way to investigate complex dynamics due to variation of economic interest. Furthermore, a discrete time delay, which represents gestation delay of the predator population, is incorporated into system (3). Consequently, compared with the work done in [12], we can investigate combined dynamic effects of time delay and additive Allee effect on population dynamics by analyzing the local stability and bifurcation phenomenon of system (3) in this paper, which makes this paper have some new and positive features.
Notes
Acknowledgements
Authors would like to express their gratitude to the editor and anonymous reviewers for valuable comments and suggestions, and for the time and efforts they have spent in the review. Without the expert comments made by the editor and anonymous reviewers, the paper would not be of this quality.
Availability of data and materials
All authors of this article declare that all parameter values and data utilized in the numerical simulation section of this paper are taken from the numerical simulation section in Ref. [12], and interested readers can access all parameter values and data in the the numerical simulation section in Ref. [12].
Authors’ information
All email addresses of authors are as follows: (Chao Liu), (Luping Wang), (Na Lu), (Longfei Yu).
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Funding
This work is supported by the National Natural Science Foundation of China, grant No. 61673099, Research Program for Liaoning Excellent Talents in University, grant No. LJQ2014027, Hebei Province Natural Science Foundation, grant No. F2015501047, and Fundamental Research Funds for the Central Universities, grant No. N162304006.
Competing interests
The authors declare that they have no competing interests. All authors of this article declare that there is no conflict of interests regarding the publication of this article. We have no proprietary, financial, professional, or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or review of this article.
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