Persistence
In biological sense, persistence means the long term survival of all
populations whatever may be the initial populations. Geometrically, it
means the existence of a region in the phase space at a non-zero
distance from the boundary in which all species enter and must lie
ultimately.
Now we state a result establishing the uniform persistence of system
(2).
Theorem 2. Suppose\(E_{12},\ E_{13}\ \mathrm{\text{and\ }}\ E_{23}\ exist.\ Further\ suppose\ that\ d<\ c_{1}\varepsilon{\overset{\overline{}}{x}}_{1}+c_{2}\mu{\overset{\overline{}}{x}}_{2},\)
\(\ \frac{r_{2}}{1+k_{2}\hat{y}}>\ \beta{\hat{x}}_{1}+\mu\hat{y}\)and\(\mathrm{\ }\frac{r_{1}}{1+k_{1}\tilde{y}}>\ \alpha{\tilde{x}}_{2}+\varepsilon\tilde{y}\ \)then system (2) is uniformly persistent.
Proof . We shall prove the theorem by using the idea of average
Lyapunov function [9].
Consider the average Lyapunov function of the form :\(H\left(x\right)={x_{1}}^{m_{1}}{x_{2}}^{m_{2}}y^{m_{3}}\), where
each \(m_{i},\ i=1,2,3\) is assumed positive. In the interior of\(\mathbb{R}_{+}^{3},\) one has
\(\frac{1}{H(x)}\frac{dH(x)}{\text{dt}}=\psi\left(x\right)=\frac{m_{1}}{x_{1}}\frac{dx_{1}}{\text{dt}}+\frac{m_{2}}{x_{2}}\frac{dx_{2}}{\text{dt}}+\frac{m_{3}}{y}\frac{\text{dy}}{\text{dt}}\)
\({=m}_{1}\left(\frac{r_{1}}{1+k_{1}y}-x_{1}-\alpha x_{2}-\varepsilon y\right)+m_{2}\left(\frac{r_{2}}{1+k_{2}y}-\beta x_{1}-x_{2}-\mu y\right)+m_{3}\left(-d+c_{1}\varepsilon x_{1}+c_{2}\mu x_{2}-hy\right).\)(3)
We have to show \(\psi\left(x\right)>0\) for all\(\text{xϵ\ bd}\mathbb{R}_{+}^{3}\) , for a suitable choice
of\(m_{1},\ \text{\ m}_{2},\ \ m_{3}>0,\) to prove uniform persistence
of system (2). That is one has to fulfil the following conditions
corresponding to the boundary equilibria\(E_{0},\ \ E_{1},\text{\ E}_{2},\ \ E_{12},\ \text{\ E}_{13},\ \text{\ E}_{23}\ \)only
as there are no periodic orbits in the\(x_{1}-x_{2},\ x_{1}-y\ \mathrm{\text{and\ }}x_{2}-y\) plane
respectively.
\(E_{0}\ :\ m_{1}r_{1}+m_{2}r_{2}-m_{3}d>0\) (4)
\(E_{1}\ :\ m_{2}\left(r_{2}-\beta r_{1}\right)+m_{3}(-d+c_{1}\varepsilon r_{1})>0\)(5)
\(E_{2}\ :\ m_{1}\left(r_{1}-\alpha r_{2}\right)+m_{3}(-d+c_{2}\mu r_{2})>0\)(6)
\(E_{12}\) :\(m_{3}\mathrm{(}\ c_{1}\varepsilon{\overset{\overline{}}{x}}_{1}+c_{2}\mu{\overset{\overline{}}{x}}_{2}-d)>0\)(7)
\(E_{13}\ :\ \ m_{2}\left(\ \frac{r_{2}}{1+k_{2}\hat{y}}-\ \beta{\hat{x}}_{1}-\mu\hat{y}\right)>0\)(8)
\(E_{23}\ :\ \ m_{1}\left(\frac{r_{1}}{1+k_{1}\tilde{y}}-\ \alpha{\tilde{x}}_{2}-\varepsilon\tilde{y}\text{\ \ \ }\right)>0\)(9)
Since\(d<\ c_{1}\varepsilon{\overset{\overline{}}{x}}_{1}+c_{2}\mu{\overset{\overline{}}{x}}_{2}\), \(\ \frac{r_{2}}{1+k_{2}\hat{y}}>\ \beta{\hat{x}}_{1}+\mu\hat{y}\)and\(\mathrm{\ }\frac{r_{1}}{1+k_{1}\tilde{y}}>\ \alpha{\tilde{x}}_{2}+\varepsilon\tilde{y}\ \)positivity of (7), (8) and (9) is obvious. Again existence of\(E_{13}\ \mathrm{\text{and\ }}E_{23\ }\mathrm{\text{implies\ that}}\)\(d<c_{1}\varepsilon r_{1}\ \mathrm{\text{and}}\ c_{2}\mu r_{2}\text{\ .}\)There are two alternative conditions for existence of \(E_{12}\) e.
g.,\(\ \left(i\right)\ \alpha<\frac{r_{1}}{r_{2}}<\ \frac{1}{\beta}\ \mathrm{or\ (ii)\ }\alpha>\frac{r_{1}}{r_{2}}>\ \frac{1}{\beta}\ \).
Condition (i) implies that\(r_{1}-\alpha r_{2}>0\ \text{and\ }r_{2}-\beta r_{1}>0\). In
this case, positivity of (5) and (6) are obvious and positivity of (4)
will follow by the suitable choice of\(m_{1},\ \ m_{2}\ \text{and\ }m_{3}\). Condition (ii) implies that\(r_{1}-\alpha r_{2}<0\ \text{and\ }r_{2}-\beta r_{1}<0\). To
show positivity of (4), (5) and(6), we have to choose \(m_{3}\) as
max\(\{\frac{m_{2}(\beta r_{1}-r_{2})}{-d+c_{1}\varepsilon r_{1}},\ \frac{m_{1}(\alpha r_{2}-r_{1})}{-d+c_{2}\mu r_{2}}\}<m_{3}<\frac{m_{1}r_{1}+m_{2}r_{2}}{d}\).
So in any case, positivity of (4), (5) and (6) will follow for suitable
choice of \(m_{i},\ i=1,\ 2,\ 3\). This completes the proof.
Now system (2) ensures uniform persistence provided that the conditions
of Theorem 2 are satisfied. Further, it is proved in [10], uniform
persistence implies the existence of an interior equilibrium point.
Hence \(E^{*}=(x_{1}^{*},\ x_{2}^{*},\ y^{*})\) exists; that is in
effect Theorem 2 implies that \(E^{*}\) exists. There may exist multiple
coexistence equilibrium point which are not investigated due to
complexity of the system.
Theorem 3. Suppose
all the conditions of Theorem 2 be satisfied. Then the interior
equilibrium point \(E^{*}\) of system (2) is locally
asymptotically stable if\(4p_{1}p_{2}>{(p_{1}\alpha+p_{2}\beta)}^{2}\ \)where\(p_{1}=\frac{c_{1}\varepsilon{(1+k_{1}y^{*})}^{2}}{r_{1}k_{1}}\text{\ and\ }p_{2}=\frac{c_{2}\mu{(1+k_{2}y^{*})}^{2}}{r_{2}k_{2}}.\)
Proof. Consider the positive definite function
\(V\left(t\right)={p_{1}(x}_{1}-x_{1}^{*}-x_{1}^{*}\ln\frac{x_{1}}{x_{1}^{*}})+p_{2}(x_{2}-x_{2}^{*}-x_{2}^{*}\ln\frac{x_{2}}{x_{2}^{*}})+\left(y-y^{*}-y^{*}\ln\frac{y}{y^{*}}\right)\).
The time derivative along the solution of system (2), can be obtained as
\(\frac{\text{dV}}{\text{dt}}=p_{1}\left(x_{1}-x_{1}^{*}\right)\left\{\frac{r_{1}}{1+k_{1}y}-x_{1}-\alpha x_{2}-\varepsilon y\right\}+p_{2}\left(x_{2}-x_{2}^{*}\right)\left(\frac{r_{2}}{1+k_{2}y}-\beta x_{1}-x_{2}-\mu y\right)+\)
\(\ \ \ \ \ \ \ \ \ \ \ (y-y^{*})(-d+c_{1}\epsilon x_{1}+c_{2}\mu x_{2}-hy)\).
We expand \(\frac{\text{dV}}{\text{dt}}\) about\((x_{1}^{*},\ x_{2}^{*},\ y^{*})\) and after some algebraic
calculations get
\(\frac{\text{dV}}{\text{dt}}=-p_{1}{(x_{1}-x_{1}^{*})}^{2}-\left(p_{1}\alpha+p_{2}\beta\right)\left(x_{1}-x_{1}^{*}\right)\left(x_{2}-x_{2}^{*}\right)-p_{2}{(x_{2}-x_{2}^{*})}^{2}-h{(y-y^{*})}^{2}+\mathrm{\text{H.\ O.\ T}}\)
where H. O. T stands for terms that are cubic or higher orders.
Now \(\frac{\text{dV}}{\text{dt}}\ \leq 0\) if\(4p_{1}p_{2}>{(p_{1}\alpha+p_{2}\beta)}^{2}\). ConsequentlyV is a Lyapunov function and hence the theorem follows.
Remark. One can also determine the local stability of \(E^{*}\)by using the Routh-Hurwitz criterion.
The characteristic equation about\(E^{*}\)is
\(\lambda^{3}+a_{1}\lambda^{2}+a_{2}\lambda+a_{3}=0\) (10)
where
\(\text{\ \ }a_{1}=x_{1}^{*}+x_{2}^{*}+hy^{*},\)
\(\text{\ \ }a_{2}=x_{1}^{*}x_{2}^{*}\left(1-\alpha\beta\right)+hy^{*}\left(x_{1}^{*}+x_{2}^{*}\right)+\left(\frac{r_{1}k_{1}}{\left(1+k_{1}y^{*}\right)^{2}}+\varepsilon\right)c_{1}\varepsilon x_{1}^{*}y^{*}+(\frac{r_{2}k_{2}}{{(1+k_{2}y^{*})}^{2}}+\mu)c_{2}\mu x_{2}^{*}y^{*},\ \)
\(\text{\ \ \ a}_{3}=x_{1}^{*}x_{2}^{*}y^{*}\{h\left(1-\alpha\beta\right)+\left(c_{2}\mu-\alpha c_{1}\varepsilon\right)\left(\frac{r_{2}k_{2}}{\left(1+k_{2}y^{*}\right)^{2}}+\mu\right)+\left(c_{1}\varepsilon-\beta c_{2}\mu\right)\left(\frac{r_{1}k_{1}}{\left(1+k_{1}y^{*}\right)^{2}}+\varepsilon\right)\}.\)
Clearly\(a_{1}>0.\ \text{\ If\ }\ a_{2}>0,\ a_{3}>0\ \mathrm{\text{and}}\ a_{1}a_{2}>\ a_{3}\)then \(E^{*}\) is locally asymptotically stable follows from
Routh-Hurwitz criterion.
Theorem 4 . Suppose
that\(4c_{1}c_{2}>{(c_{1}\alpha+c_{2}\beta)}^{2}\)and det A > 0 where A is defined in the proof. Then\(E^{*}\) is globally asymptotically stable.
Proof. Consider the
following positive definite function about\(E^{*}.\)
\(V\left(t\right)={c_{1}(x}_{1}-x_{1}^{*}-x_{1}^{*}\ln\frac{x_{1}}{x_{1}^{*}})+c_{2}(x_{2}-x_{2}^{*}-x_{2}^{*}\ln\frac{x_{2}}{x_{2}^{*}})+\left(y-y^{*}-y^{*}\ln\frac{y}{y^{*}}\right)\).
Differentiating \(V\) with respect to \(t\)along the solution of system (2),
we get
\begin{equation}
\frac{\text{dV}}{\text{dt}}=c_{1}\left(x_{1}-x_{1}^{*}\right)\left\{\frac{r_{1}}{1+k_{1}y}-x_{1}-\alpha x_{2}-\varepsilon y\right\}+c_{2}\left(x_{2}-x_{2}^{*}\right)\left(\frac{r_{2}}{1+k_{2}y}-\beta x_{1}-x_{2}-\mu y\right)+\nonumber \\
\end{equation}\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (y-y^{*})(-d+c_{1}\epsilon x_{1}+c_{2}\mu x_{2}-hy)\)
\begin{equation}
=c_{1}\left(x_{1}-x_{1}^{*}\right)\left\{\frac{r_{1}k_{1}\left(y^{*}-y\right)}{\left(1+k_{1}y\right)\left(1+k_{1}y^{*}\right)}-{(x}_{1}-x_{1}^{*})-\alpha{(x}_{2}-x_{2}^{*})\right\}\nonumber \\
\end{equation}\(=c_{1}\left(x_{1}-x_{1}^{*}\right)\left\{\frac{r_{1}k_{1}\left(y^{*}-y\right)}{\left(1+k_{1}y\right)\left(1+k_{1}y^{*}\right)}-{(x}_{1}-x_{1}^{*})-\alpha{(x}_{2}-x_{2}^{*})\right\}-c_{2}\left(x_{2}-x_{2}^{*}\right)\left\{\frac{r_{2}k_{2}\left(y^{*}-y\right)}{\left(1+k_{2}y\right)\left(1+k_{2}y^{*}\right)}-\beta\left(x_{1}-x_{1}^{*}\right)-\left(x_{2}-x_{2}^{*}\right)\right\}-h{(y-y^{*})}^{2}\)
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\leq-c_{1}\left(x_{1}-x_{1}^{*}\right)^{2}+\left(c_{1}\alpha+c_{2}\beta\right)\left|\left(x_{1}-x_{1}^{*}\right)\right|\left|\left(x_{2}-x_{2}^{*}\right)\right|-c_{2}\left(x_{2}-x_{2}^{*}\right)^{2}-h{(y-y^{*})}^{2}+\frac{c_{1}r_{1}k_{1}}{{(1+k_{1}y^{*})}^{2}}\left|x_{1}-x_{1}^{*}\right|\left|y-y^{*}\right|+\frac{c_{2}r_{2}k_{2}}{{(1+k_{2}y^{*})}^{2}}|x_{2}-x_{2}^{*}||y-y^{*}|\)
Clearly \(\dot{V}\) is negative definite if the matrixA defined below is positive definite.
\begin{equation}
A=\begin{pmatrix}c_{1}&-\frac{1}{2}(c_{1}\alpha+c_{2}\beta)&-\frac{c_{1}r_{1}k_{1}}{2{(1+k_{1}y^{*})}^{2}}\\
-\frac{1}{2}(c_{1}\alpha+c_{2}\beta)&c_{2}&-\frac{c_{2}r_{2}k_{2}}{2{(1+k_{2}y^{*})}^{2}}\\
-\frac{c_{1}r_{1}k_{1}}{2{(1+k_{1}y^{*})}^{2}}&-\frac{c_{2}r_{2}k_{2}}{2{(1+k_{2}y^{*})}^{2}}&h\\
\end{pmatrix}\nonumber \\
\end{equation}Thus the condition of the theorem implies thatA is positive definite and consequentlyV is a Lyapunov
function and hence the theorem follows.