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A Predator-Prey Model in the Chemostat with Time Delay
The aim of this paper is to study the dynamics of predator-prey interaction in a chemostat to determine whether including a discrete delay to model the time between the capture of the prey and its conversion to viable biomass can introduce oscillatory dynamics even though there is a globally asymptotically stable equilibrium when the delay is ignored. Hence, Holling type I response functions are chosen so that no oscillatory behavior is possible when there is no delay. It is proven that unlike the analogous model for competition, as the parameter modeling the delay is increased, Hopf bifurcations can occur.
The chemostat, also known as a continuous stir tank reactor (CSTR) in the engineering literature, is a basic piece of laboratory apparatus used for the continuous culture of microorganisms. It has potential applications for such processes as wastewater decomposition and water purification. Some ecologists consider it a lake in a laboratory. It can be thought of as three vessels, the feed bottle that contains fresh medium with all the necessary nutrients, the growth chamber where the microorganisms interact, and the collection vessel. The fresh medium from the feed bottle is continuously added to the growth chamber. The growth chamber is well stirred and its contents are then removed to the collection vessel at a rate that maintains constant volume. For a detailed description of the importance of the chemostat and its application in biology and ecology, one can refer to [1, 2].
The following system describes a food chain in the chemostat where a predator population feeds on a prey population of microorganisms that in turn consumes a nonreproducing nutrient that is assumed to be growth limiting at low concentrations
Here represents the concentration of the growth limiting nutrient, the density of the prey population, and the density of the predator population. Parameter denotes the concentration of the growth limiting nutrient in the feed vessel, the dilution rate, the growth yield constant, the sum of the dilution rate and the natural species specific death rate of the prey (predator) population, respectively. Here denotes the functional response of the prey population on the nutrient and denotes the functional response of the predator on the prey.
Butler et al.  considered the coexistence of two competing predators feeding on a single prey population growing in the chemostat. As a subsystem of their model, they studied the global stability of system (1.1) with both and taking the form of Holling type II. They proved that under certain conditions the interior equilibrium is globally asymptotically stable with respect to the interior of the positive cone. However, they also proved that for certain ranges of the parameters there is at least one nontrivial limit cycle and conjectured that the limit cycle is unique and would be a global attractor with respect to the noncritical orbits in the open positive octant. This conjecture was partially solved by Kuang . He showed that there is a range of parameters for which a unique periodic orbit exists and roughly located the position of the limit cycle.
Bulter and Wolkowicz  studied predator mediated coexistence in the chemostat assuming . Model (1.1) was studied as a submodel. For general monotone response functions, Bulter and Wolkowicz showed that (1.1) is uniformly persistent if the sum of the break even concentrations of substrate and prey is less than the input rate of the nutrient . However they showed that it is necessary to specify the form of the response functions in order to discuss the global dynamics of the model. If is modelled by Holling type I or II and by Holling type I, Bulter and Wolkowicz proved that (1.1) could have up to three equilibrium points and that there is a transfer of global stability from one equilibrium point to another as different parameters are varied making conditions favorable enough for a new population to survive. In this case, there are no periodic solutions. However, even if is given by Holling type I, if is given by Holling type II, they showed that a Hopf bifurcation can occur in (1.1), and numerical simulations indicated that the bifurcating periodic solution was asymptotically stable.
We include a time delay in (1.1) to model the time between the capture of the prey and its conversion to viable biomass. Our aim is to show that such a delay can induce nontrivial periodic solutions in a model where there is always a globally asymptotically stable equilibrium when delay is ignored, and hence no such periodic solutions are possible otherwise. For this reason we select the response functions of the simplest form; that is, we choose the Holling type I form for both and , so that (1.2) always has a globally asymptotically stable equilibrium when the conversion process is assumed to occur instantaneously. It is interesting to note that in the analogous model of competition between two species in the chemostat, delay cannot induce oscillatory behavior for any reasonable monotone response functions (see Wolkowicz and Xia ).
With delay modelling the time required for the predator to process the prey after it has been captured, the model is given by
Here variables , , , and parameters , , , , , , and have the same interpretation as for model (1.1). Note therefore that , and . The additional parameter is a nonnegative constant modelling the time required for the conversion process. Hence, represents the concentration of the predator population in the growth chamber at time that were available at time to capture prey and were able to avoid death and washout during the units of time required to process the captured prey.
We analyze the stability of each equilibrium and prove that the coexistence equilibrium can undergo Hopf bifurcations. Numerical simulations appear to show that (1.2) can have a stable periodic solution bifurcating from the coexistence equilibrium as the delay parameter increases from zero. This periodic orbit can then disappear through a secondary Hopf bifurcation as the delay parameter increases further.
2. Scaling of the Model and Existence of Solutions
Suppose that functions and are of Holling type I form, that is, () and (). System (1.2) reduces to
Introducing the following change of variables gives:
With this change of variables, omitting the ’s for convenience, system (2.1) becomes
Let . Model (2.3) reduces to a special case of the model considered in . If , the model has only one equilibrium point and it is globally asymptotically stable. If and , the model has a second equilibrium point and it is globally asymptotically stable. When , the model has a third equilibrium point and it is the global attractor. Therefore, model (2.3) has no periodic solutions when the time delay is ignored. If is of Holling type II form, Butler and Wolkowicz  proved that a Hopf bifurcation is possible resulting in a periodic solution for a certain range of parameter values. We emphasize again here, that it is for this reason that in this paper we restrict our attention to the simplest case for both response functions, that is, Holling type I, in order to see whether delay can be responsible for periodic solutions in (1.2).
Theorem 2.1. Assuming , then there exists a unique solution of (2.3) passing through with , and for . The solution is bounded. In particular, given any , for all sufficiently large .
Proof. For , one has , and . System (2.3) becomes
a system of nonautonomous ordinary differential equations with initial conditions , , and . Since the right-hand side of (2.4) is differentiable in both and , by Theorems , , and Corollary in Miller and Michel , there exists a unique solution defined on satisfying (2.4). By using the method of steps in Bellman and Cooke , it can be shown that the solution through is defined for all .
Now we prove for all . From the first equation of (2.3),
Proceed using the method of contradiction. Suppose that there exists a first such that and for . Then . But from the first equation of (2.3) a contradiction.
To prove for , assume there is a first such that , and for . Divide both sides of the second equation of (2.3) by and integrate from to , to obtain
To show that is positive on , suppose that there exists such that and for . Then From the third equation of (2.3), we have
To prove the boundedness of solutions, define
It follows that where the first inequality holds since , , and . It follows that Therefore, the solution is bounded, and given any , for all sufficiently large .
3. Equilibria and Stability
Model (2.3) has three equilibrium points: , and
We call the washout equilibrium, the single species equilibrium, and the coexistence equilibrium. For the sake of biological significance, exists (distinct from ) if and only if its third coordinate , that is, , or equivalently, lies between and , where
Note that if , the equilibrium does not exist for any (0), and if , then .
The linearization of (2.3) about an equilibrium is given by
The associated characteristic equation is given by
Direct calculation of the left-hand side of (3.4) gives
For convenience, define
Theorem 3.1. Equilibrium is stable if and unstable if .
Proof. Evaluating the characteristic equation at gives The eigenvalues and are both negative. The third eigenvalue is . Therefore the equilibrium is stable if and unstable if .
Remark 3.2. If , then there is only one equilibrium, . If , equilibrium also exists.
Lemma 3.3. Assume . The characteristic equation evaluated at has two negative eigenvalues, and the remaining eigenvalues are solutions of In addition, the characteristic equation evaluated at has zero as an eigenvalue if and only if .
Proof. Assume . Equilibrium exists. Consider the characteristic equation at . Since at ,
where and . Therefore, and have negative real parts. The rest of the eigenvalues are roots of (3.8).
Assuming that is a root of (3.8), we have
Solving for gives
Theorem 3.4. Assume that , , , , and so that . Equilibrium is locally asymptotically stable if and unstable if . If , then equilibrium is globally asymptotically stable for .
Proof. Assume that . Assumptions , , and imply , or equivalently . By Lemma 3.3, to prove that equilibrium is locally asymptotically stable, one only needs to show that (3.8) admits no root with nonnegative real part.
Consider the real roots of (3.8) first. Note that . Equation (3.8) has no solution for . Otherwise the left-hand side would be less than zero, but the right-hand side would be greater than zero. Assume . The left-hand side of (3.8) is a monotone increasing function in both and , takes value at , and goes to positive infinity as or . By Lemma 3.3, when , then is a solution of (3.8). Thus for , any real root of (3.8) must satisfy .
For any , we have and . Therefore there exists at least one such that is a solution of (3.8). Equilibrium is unstable if .
In what follows, we prove that if all complex eigenvalues of (3.8) have negative real parts. Suppose that is a solution of (3.8). Using the Euler formula, we have
Equating the real parts and imaginary parts of the equation, we have Squaring both equations, adding, and taking the square root on both sides give The left-hand side of (3.14) is monotonically increasing in , , and provided that . Since (3.14) has solution , at , any roots of (3.14) must satisfy since . Hence . Therefore (3.8) has no complex eigenvalue with nonnegative real part and so is locally asymptotically stable for .
Assume that . Now we prove that is globally asymptotically stable when , or equivalently . In this case, choose small enough such that By Theorem 2.1, for such , there exists a so that for . Hence, for , In Example of Kuang ([10, page 32]), choose , , , and . We obtain . Therefore as . Let . Noting , from (2.3), we have . Multiply by the integrating factor , . Integrating both sides from to gives
If , then . Therefore . If , by L'Hôspital's rule, since is bounded and . It again follows that . Hence
We show that and . First assume that the limits exist, that is, and . From (2.3), we know that and are uniformly continuous since , , and are bounded. By Theorem A.3, it follows that and . Note that . Letting in (2.3) gives
Either or . Assume that , that is, and . Note that . There exists such that . For such , there exists a sufficiently large so that and . Recalling that , by (2.3) for all sufficiently large . Therefore it is impossible for to approach from above giving a contradiction. Therefore, we must have .
Now suppose that the limits do not exist. In particular if does not converge, then let and . By Lemma A.2 in the appendix, there exists and such that
From (2.3), Noting that , we have . Since , . By (3.17), . Therefore . Similarly we can show that . This implies that , a contradiction.
Since converges and converges, then must also converge. Hence and . It follows that is globally asymptotically stable.
4. Hopf Bifurcations at Assuming
Now consider the stability of . The characteristic equation at is
By assumption , and so
The characteristic equation at has one eigenvalue equal to and the others are given by solutions of the equation
Lemma 4.1. Assuming , , and so that , then has no zero eigenvalue for .
Proof. Assume that . By the method of contradiction, suppose that there exists a zero root of (4.4). Therefore Noting that if and only if , for any , a contradiction.
Lemma 4.2. Assume , , . Equilibrium is asymptotically stable when .
Proof. For , (4.4) reduces to Both coefficients are positive, since and implies . Therefore, all the roots of the characteristic equation have negative real parts.
Lemma 4.3. As is increased from , a root of (4.4) with positive real part can only appear if a root with negative real part crosses the imaginary axis.
Proof. Taking and in Kuang [10, Theorem , page 66] gives
Therefore, no root of (4.4) with positive real part can enter from infinity as increases from . Hence roots with positive real part can only appear by crossing the imaginary axis.
For , assuming () is a root of ,
Substituting into (4.10) gives Separating the real and imaginary parts, we obtain Solving for and gives Noting , squaring both sides of equations (4.13), adding, and rearranging gives Solving for , we obtain two roots and :
Define conditions () and () as follows:
Lemma 4.4. If () holds for all in some interval , then (4.14) has two positive roots for all with when all the inequalities in () are strict. If () holds for all in some interval , then (4.14) has only one positive root, for all . If no interval exists where either () or () holds, then there are no positive real roots of (4.14).
Define the interval
When the end points of are real and , define
We prove that holds for any .
If , then . It follows that
Theorem 4.5. Assume and , then is not empty, and for any , but , condition holds and . If , then .
Proof. For any , we have , and therefore
Therefore, . Since , it follows that
From , we have . Therefore,
and so is not empty. Noting and , for any , but , we have .
In what follows, we intend to show that for any such , condition holds. From (4.3),
Since , to show that the first inequality in holds, it suffices to show that the factor on the right-hand side of the above expression is positive. Since , , and Since for , For any , Hence, Next consider the second inequality in . For , since , . Therefore, . For Finally, where Since , It follows that . Again noting that ,