In the section, the local stability of differential-algebraic system (2) with discrete time delay τ at the boundary equilibrium is investigated in the case of economic equilibrium. Furthermore, the phenomenon of Hopf bifurcation at one boundary equilibrium is also investigated.
When \(m=0\), (2) can be written as
$$ \textstyle\begin{cases} \frac{{dx_{1} ( t ) }}{{dt}} = ax_{2} ( t ) - {r_{1}}x_{1} ( t ) - \beta x_{1} ( t ), \\ \frac{{dx_{2} ( t ) }}{{dt}} = \beta x_{1} ( t ) - {r_{2}}x_{2} ( t ) - {s_{1}}x_{2}^{2} ( t ) - {\beta_{1}}x_{2} ( t ) y ( t ), \\ \frac{{dy ( t ) }}{{dt}} = {\beta_{1}}x_{2} ( {t - \tau } ) y ( {t - \tau } ) - {r_{3}}y ( t ) - {s_{2}}{y^{2}} ( t ) - E ( t ) y ( t ), \\ 0 = E ( t ) ( {py ( t ) - c} ). \end{cases} $$
(4)
It is clear that (4) has three boundary equilibrium points: \({P_{1}} ( {0,0,0,0} ) \), \(P_{2} (x_{11},x_{21}, 0,0 ) \), \({P_{2}} ( {{x_{13}},x_{23}, {y_{3}},0} ) \), where \({x_{11}} = \frac{a}{{{r_{1}} + \beta }} {x_{21}}\), \({x_{21}} = \frac{{\beta a - {r_{2}} ( {{r_{1}} + \beta } ) }}{{ ( {{r_{1}} + \beta } ) {s_{1}}}}\), \({x_{13}} = \frac{a}{{{r_{1}} + \beta }}x_{23}\), \(x_{23} = \frac{ {{s_{2}} ( {a\beta - {r_{1}}{r_{2}} - {r_{2}}\beta } ) + \beta {r_{3}} ( {{r_{1}} + \beta } ) }}{{ ( {{r_{1}} + \beta } ) ( {{s_{1}}{s_{2}} + \beta {\beta_{1}}} ) }}\).
For \({P_{2}} ( {{x_{11}},{x_{21}},0,0} ) \), in order to guarantee that the juveniles and adults of the prey population all exist, the following inequality needs to be satisfied: \(\beta a - {r_{2}} ( {{r_{1}} + \beta } ) > 0\). Similarly, for \({P_{3}} ( {{x_{13}},x_{23},{y_{3}},0} ) \), the following inequality is satisfied: \({\beta_{1}} ( {a\beta - {r_{1}}{r_{2}} - {r_{2}}\beta } ) - {r_{3}} ( {{r_{1}} + \beta } ) {s_{1}} > 0\). Next, we can consider the stability in the neighborhood of each boundary equilibrium.
Theorem 4.1
Equation (4) is unstable at the equilibrium point
\({P_{1}} ( {0,0,0,0} ) \).
Proof
The Jacobian matrix of model (4) is given by
$$\begin{aligned} J &= {D_{X}}F - {D_{E}}F{ ( {{D_{E}}G} ) ^{ - 1}} {D_{X}}G \\ &= \left [ { \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} { - {r_{1}} - \beta } & a & 0 \\ \beta & { - {r_{2}} - 2{s_{1}}x_{2} - {\beta_{1}}y} & { - {\beta _{1}}x_{2}} \\ 0 & {{\beta_{1}}y{e^{ - \lambda \tau }}} & {{\beta_{1}}x_{2}{e^{ - \lambda \tau }} - {r_{3}} - 2{s_{2}}y - E + \frac{{pEy}}{{py - c}}} \end{array}\displaystyle } \right ] . \end{aligned}$$
Then the characteristic polynomial at \(P_{1}\) is
$$ \det ( {\lambda A - {J_{{P_{1}}}}} ) = ( {\lambda + {r_{3}}} ) \bigl( {{\lambda^{2}} + ( {{r_{1}} + {r_{2}} + \beta } ) \lambda + ( {{r_{1}} + \beta } ) {r_{2}} - a \beta } \bigr) = 0, $$
where
$$ A = \left [ { \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\displaystyle } \right ] . $$
Based on the condition of the existence of the adult prey population: \({r_{1}} + {r_{2}} + \beta > 0\), \(( {{r_{1}} + \beta } ) {r_{2}} - a\beta < 0\). It can be judged that \({P_{1}} ( {0,0,0,0} ) \) is an unstable equilibrium. □
Similarly, in order to study the stability of \(P_{2}\), we firstly obtain the Jacobian matrix of model (4) at \(P_{2}\):
$$ {J_{{P_{2}}}} = \left [ { \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} { - {r_{1}} - \beta } & a & 0 \\ \beta & { - {r_{2}} - 2{s_{1}}{x_{21}}} & { - {\beta_{1}}{x_{21}}} \\ 0 & 0 & {{\beta_{1}}{x_{21}}{e^{ - \lambda \tau }} - {r_{3}}} \end{array}\displaystyle } \right ] . $$
Then the characteristic polynomial at \(P_{2}\) is
$$\begin{aligned} & \det ( {\lambda A - {J_{{P_{2}}}}} ) \\ & \quad = \bigl( { \lambda + {r_{3}} - {\beta_{1}} {x_{21}} {e^{ - \lambda \tau }}} \bigr) \bigl( {{\lambda^{2}} + ( {{r_{1}} + {r_{2}} + \beta + 2{s_{1}} {x_{21}}} ) \lambda + ( {{r_{1}} + \beta } ) ( {{r_{2}} + 2 {s_{1}} {x_{21}}} ) - a\beta } \bigr) \\ &\quad = 0. \end{aligned}$$
Based on the above analysis, we obtain the following:
$$\begin{aligned}& ( {{r_{1}} + \beta } ) ( {{r_{2}} + 2{s_{1}} {x_{21}}} ) - a\beta = a\beta - {r_{2}} ( {{r_{1}} + \beta } ) > 0, \\& {r_{1}} + {r_{2}} + \beta + 2{s_{1}} {x_{21}} = a\beta - {r_{2}} ( {{r_{1}} + \beta } ) > 0. \end{aligned}$$
So the other characteristic of system (4) at \(P_{2}\) is determined in the following formula:
$$ \lambda + {r_{3}} - \frac{{{\beta_{1}} ( {a\beta - {r_{2}} ( {{r_{1}} + \beta } ) } ) }}{{ ( {{r_{1}} + \beta } ) {s_{1}}}}{e^{ - \lambda \tau }} = 0. $$
By the analysis of roots for the above formula, we obtain that the equilibrium point \(P_{2}\) is a stable focus or node if \({s_{1}}{r_{3}} ( {{r_{1}} + \beta } ) > {\beta_{1}} ( {a\beta - {r _{1}}{r_{2}} - {r_{2}}\beta } ) \); otherwise \(P_{2}\) is a saddle.
Similarly, the Jacobian matrix of model (4) at the equilibrium point \(P_{3}\) is given by
$$\begin{aligned}& {J_{{P_{3}}}} = \left [ { \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} { - {r_{1}} - \beta } & a & 0 \\ \beta & { - {r_{2}} - 2{s_{1}}x_{23} - {\beta_{1}}{y_{3}}} & { - {\beta_{1}}x_{23}} \\ 0 & {{\beta_{1}}{y_{3}}{e^{ - \lambda \tau }}} & {{\beta_{1}}x_{23} {e^{ - \lambda \tau }} - {r_{3}} - 2{s_{2}}{y_{3}}} \end{array}\displaystyle } \right ], \\& \textstyle\begin{array}{l} D ( {\lambda,\tau } ) = \det \left ( { \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} {\lambda + {r_{1}} + \beta } & { - a} & 0 \\ { - \beta } & {\lambda + {r_{2}} + 2{s_{1}}x_{23} + {\beta_{1}} {y_{3}}} & {{\beta_{1}}x_{23}} \\ 0 & { - {\beta_{1}}x_{23}{e^{ - \lambda \tau }}} & {\lambda + {r _{3}} + 2{s_{2}}{y_{3}} - {\beta_{1}}x_{23}{e^{ - \lambda \tau }}} \end{array}\displaystyle } \right ) \\ \hphantom{D ( {\lambda,\tau } )}= P ( \lambda ) + Q ( \lambda ) {e^{ - \lambda \tau }} = 0. \end{array}\displaystyle \end{aligned}$$
It can be simplified as follows:
$$ {\lambda^{3}} + {p_{1}} { \lambda^{2}} + {p_{2}}\lambda + {p_{3}} + \bigl( {{q_{1}} {\lambda^{2}} + {q_{2}}\lambda + {q_{3}}} \bigr) {e ^{ - \lambda \tau }} = 0, $$
(5)
where
$$\begin{aligned}& P ( \lambda ) = {\lambda^{3}} + {p_{1}} { \lambda^{2}} + {p_{2}}\lambda + {p_{3}}, \\& Q ( \lambda ) = {q_{1}} {\lambda^{2}} + {q_{2}} \lambda + {q_{3}}, \\& {p_{1}} = {r_{1}} + \beta + {r_{2}} + 2{s_{1}} x_{23} + {\beta_{1}} {y_{3}} + {r_{3}} + 2{s_{2}} {y_{3}}, \\& {p_{2}} = ( {{r_{1}} + \beta } ) ( {{r_{2}} + 2{s_{1}} x_{23} + {\beta_{1}} {y_{3}}} ) + ( {{r_{3}} + 2{s_{2}} {y_{3}}} ) ( {{r_{1}} + \beta + {r_{2}} + 2{s_{1}} x_{23} + {\beta_{1}} {y_{3}}} ) - a\beta, \\& {p_{3}} = \bigl( { ( {{r_{1}} + \beta } ) ( {{r_{2}} + 2 {s_{1}} x_{23} + { \beta_{1}} {y_{3}}} ) - a\beta } \bigr) ( {{r_{3}} + 2{s_{2}} {y_{3}}} ), \\& q_{1}=-\beta_{1}x_{23}, \qquad {q_{2}} = - \bigl( {{r_{2}} {\beta_{1}} x_{23} + 2{s_{1}} {\beta_{1}} x_{23}^{2} + ( {{r_{1}} + \beta } ) {\beta_{1}} x_{23}} \bigr), \\& {q_{3}} = a\beta {\beta_{1}} x_{23} - ( {{r_{1}} + \beta } ) {r_{2}} {\beta_{1}} x_{23} - 2{s_{1}} ( {{r_{1}} + \beta } ) { \beta_{1}} x_{23}^{2}. \end{aligned}$$
It is assumed that for some values of \(\tau > 0\), there exists a real such that \(\lambda = i\omega \) is a root of the characteristic equation (5). Now, substituting \(\lambda = i\omega \) into equation (5), we have
$$ - i{\omega^{3}} - {n_{1}} { \omega^{2}} + i{p_{2}}\omega + {n_{3}} + \bigl( { - {n_{4}} {\omega^{2}} + i{n_{5}}\omega + {n_{6}}} \bigr) {e^{ - i\omega \tau }} = 0. $$
Then, by separating real and imaginary parts of \(D=0\), we obtain that
$$\begin{aligned}& {\omega^{3}} - {p_{2}}\omega = {q_{2}}\omega \cos ( {\omega \tau } ) - \bigl( {{q_{3}} - {q_{1}} {\omega^{2}}} \bigr) \sin ( {\omega \tau } ), \end{aligned}$$
(6)
$$\begin{aligned}& {p_{1}} {\omega^{2}} - {p_{3}} = \bigl( {{q_{3}} - {q_{1}} {\omega^{2}}} \bigr) \cos ( {\omega \tau } ) + {q_{2}}\omega \sin ( {\omega \tau } ) . \end{aligned}$$
(7)
By squaring and adding (6) and (7), it can be shown that
$$\begin{aligned}& {\omega^{6}} + \bigl( {{p_{1}}^{2} - 2{p_{2}} - {q_{1}}^{2}} \bigr) { \omega^{4}} + \bigl( {{p_{2}}^{2} - {q_{2}}^{2} - 2{p_{1}} {p_{3}} + 2 {q_{1}} {q_{3}}} \bigr) {\omega^{2}} + {p_{3}}^{2} - {q_{3}}^{2} = 0, \\& {\omega^{6}} + {C_{1}} { \omega^{4}} + {C_{2}} {\omega^{2}} + {C_{3}} = 0, \end{aligned}$$
(8)
where parameters \(C_{1}\), \(C_{2}\), \(C_{3}\) can be expressed as follows:
$$\begin{aligned}& {C_{1}} = {p_{1}}^{2} - 2{p_{2}} - {q_{1}}^{2}, \\& {C_{2}} = {p_{2}}^{2} - 2{p_{1}} {p_{3}} - {q_{2}}^{2} + 2{q_{1}} {q _{3}}, \\& {C_{3}} = {p_{3}}^{2} - {q_{3}}^{2}. \end{aligned}$$
According to the Routh-Hurwitz criteria [20], in order to guarantee equation (8) has at least one real root \({\omega _{0}}\), \(C_{1}\), \(C_{2}\), \(C_{3}\) need to satisfy any one of which is less than zero. A simple assumption that equation (8) has a positive real root \({\omega_{0}}\) is \({C_{3}} < 0\). Hence, under the assumption, equation (8) will have a pair of purely imaginary roots of the form \(\pm i{\omega_{0}}\).
By computing, it can be obtained that \({\tau_{n}}\) corresponding to \({\omega_{0}}\) is as follows:
$$ {\tau_{n}} = \frac{1}{{{\omega_{0}}}}\arccos \biggl( { \frac{{ ( {{p_{1}}{\omega_{0}}^{2} - {p_{3}}} ) ( {{q_{3}} - {q_{1}} {\omega_{0}}^{2}} ) + {q_{2}}{\omega_{0}} ( {{\omega_{0}} ^{3} - {p_{2}}{\omega_{0}}} ) }}{{{{ ( {{q_{3}} - {q_{1}} {\omega_{0}}^{2}} ) }^{2}} + {{ ( {{q_{2}}{\omega_{0}}} ) } ^{2}}}}} \biggr) + \frac{{2n\pi }}{{{\omega_{0}}}}, $$
(9)
where \(n = 0,1,2, \ldots \) .
Based on Butler’s lemma in [21], the conclusion can be expressed as follows: system (4) is stable at the boundary equilibrium point \(P_{3}\) for \(\tau < {\tau_{0}}\).
Theorem 4.2
System (4) undergoes a Hopf bifurcation at the equilibrium point
\(P_{3}\)
when
\(\tau = {\tau_{0}}\)
if the inequality
\({p_{1}}^{2} - 2 {p_{2}} - {q_{1}}^{2} > 0\)
is satisfied.
Proof
Let
$$\begin{aligned} \Theta &= \operatorname{sign} { \biggl[ {\frac{{d ( {\operatorname{Re} \lambda } ) }}{{d\tau }}} \biggr] _{\lambda = i{\omega_{0}}}} \\ &= \operatorname{sign} { \biggl[ {\operatorname{Re} {{ \biggl( { \frac{{d\lambda }}{ {d\tau }}} \biggr) }^{ - 1}}} \biggr] _{\lambda = i{\omega_{0}}}}. \end{aligned}$$
By differentiating (5) with respect to τ, we can obtain
$$\begin{aligned} & \bigl( {3{\lambda^{2}} + 2{p_{1}} + {p_{2}}} \bigr) \frac{{d\lambda }}{ {d\tau }} + ( {2{q_{1}}\lambda + {q_{2}}} ) {e^{ - \lambda \tau }}\frac{{d\lambda }}{{d\tau }} - \bigl( {{q_{1}} { \lambda^{2}} + {q_{2}}\lambda + {q_{3}}} \bigr) \tau {e^{ - \lambda \tau }}\frac{ {d\lambda }}{{d\tau }} \\ &\quad = \bigl( {{q_{1}} {\lambda^{2}} + {q_{2}} \lambda + {q_{3}}} \bigr) {e^{ - \lambda \tau }}\lambda . \end{aligned}$$
Then we get
$$\begin{aligned} { \biggl( {\frac{{d\lambda }}{{d\tau }}} \biggr) ^{ - 1}} &= \frac{{3 {\lambda^{2}} + 2{p_{1}}\lambda + {p_{2}}}}{{\lambda {e^{ - \lambda \tau }} ( {{q_{1}}{\lambda^{2}} + {q_{2}}\lambda + {q_{3}}} ) }} + \frac{{2{q_{1}}\lambda + {q_{2}}\lambda }}{{\lambda ( {{q_{1}}{\lambda^{2}} + {q_{2}}\lambda + {q_{3}}} ) }} - \frac{ \tau }{\lambda } \\ &= \frac{{2{\lambda^{3}} + {p_{1}}{\lambda^{2}} - {p_{3}}}}{{ - {\lambda^{2}} ( {{\lambda^{3}} + {p_{1}}{\lambda^{2}} + {p_{2}}\lambda + {p _{3}}} ) }} + \frac{{{q_{1}}{\lambda^{2}} - {q_{3}}}}{{{\lambda ^{2}} ( {{q_{1}}{\lambda^{2}} + {q_{2}}\lambda + {q_{3}}} ) }} - \frac{\tau }{\lambda } . \end{aligned}$$
By computing the real part of \({ ( {\frac{{d\lambda }}{{d\tau }}} ) ^{ - 1}}\) for \(\lambda = i{\omega_{0}}\), we can obtain
$$\begin{aligned} & \biggl[ \operatorname{Re} \biggl(\frac{d\lambda }{d\tau } \biggr)^{ - 1} \biggr] _{\lambda = i{\omega_{0}}} \\ &\quad = { \biggl[ \operatorname{Re} \biggl( { \frac{{2{\lambda^{3}} + {p_{1}}{\lambda^{2}} - {p_{3}}}}{{ - {\lambda^{2}} ( {{\lambda ^{3}} + {p_{1}}{\lambda^{2}} + {p_{2}}\lambda + {p_{3}}} ) }}} \biggr) + \frac{{{q_{1}}{\lambda^{2}} - {q_{3}}}}{{{\lambda^{2}} ( {{q_{1}}{\lambda^{2}} + {q_{2}}\lambda + {q_{3}}} ) }} - \frac{ \tau }{\lambda } \biggr] _{ | {\lambda = i{\omega_{0}}} }} \\ &\quad = \operatorname{Re} \biggl( {\frac{{ - 2i{\omega_{0}}^{3} - {p_{1}}{\omega_{0}}^{2} - {p_{3}}}}{{{\omega_{0}}^{2} ( { - i {\omega_{0}}^{3} - {p_{1}}{\omega_{0}}^{2} + {p_{2}}i{\omega_{0}} + {p_{3}}} ) }} + \frac{{ - {q_{1}}{\omega_{0}}^{2} - {q_{3}}}}{ { - {\omega_{0}}^{2} ( { - {q_{1}}{\omega_{0}}^{2} + {q_{2}}i {\omega_{0}} + {q_{3}}} ) }}} \biggr) \\ &\quad = \frac{1}{{{\omega_{0}}^{2}}}\operatorname{Re} \biggl( {\frac{{ ( { - {p_{1}}{\omega_{0}}^{2} - {p_{3}}} ) - 2 {\omega_{0}}^{3}i}}{{ ( {{p_{3}} - {p_{1}}{\omega_{0}}^{2}} ) + ( {{p_{2}}{\omega_{0}} - {\omega_{0}}^{3}} ) i}} + \frac{ { - {q_{1}}{\omega_{0}}^{2} - {q_{3}}}}{{ ( {{q_{3}} - {q_{1}} {\omega_{0}}^{2}} ) + {q_{2}}i{\omega_{0}}}}} \biggr) \\ &\quad = \frac{1}{{{\omega_{0}}^{2}}}\operatorname{Re} \biggl[ {\frac{{2{\omega_{0}}^{6} + ( {{p_{1}}^{2} - 2{p_{2}} - {q_{1}} ^{2}} ) {\omega_{0}}^{4} - ( {{p_{3}}^{2} - {q_{3}}^{2}} ) }}{ {{{ ( {{q_{3}} - {q_{1}}{\omega_{0}}^{2}} ) }^{2}} + {{ ( {{q_{2}}{\omega_{0}}} ) }^{2}}}}} \biggr] . \end{aligned}$$
According to the assumption for value of \(C_{1}\), furthermore, based on the assumption that equation (8) has a positive real root, it follows that \({C_{3}} < 0\). Hence, the conclusion can be expressed as follows:
$$ \Theta = \operatorname{sign} { \biggl[ {\operatorname{Re} {{ \biggl( { \frac{ {d\lambda }}{{d\tau }}} \biggr) }^{ - 1}}} \biggr] _{\lambda = i{\omega _{0}}}} > 0. $$
Hence, system (4) has at least one eigenvalue with a positive real part; furthermore, system (4) undergoes a Hopf bifurcation [22] at the equilibrium point \(P_{3}\) when \(\tau =\tau_{0}\), where \(\tau_{0}\) is the bifurcation value. Thus the proof is completed. □
Remark 4.1
It follows from Theorem 4.2 that the juvenile prey, the adult prey and the predator are existent in the case of \(\tau =\tau_{0}\), while in the absence of harvested effort, system (4) will undergo a Hopf bifurcation. When the delay varies in a certain interval, system (4) is stable, which means the biological system is sustainable. But when the delay is beyond the threshold, it will become unstable.