How Can Viruses Affect the Growth of Zooplankton on Phytoplankton in a Chemostat?
Abstract
:1. Introduction
2. Mathematical Modeling
- The susceptible phytoplankton enter into the chemostat at a constant input concentration, .
- The susceptible phytoplankton become infected through direct contact with free viruses, , living in the system, with a transmission rate of .
- The zooplankton grow on susceptible and infected phytoplankton. The susceptible phytoplankton are consumed by the zooplankton Z at a rate of ; however, the infected phytoplankton are consumed by the zooplankton Z at a rate of .
- Let us denote by the virus replication factor in the infected phytoplankton , i.e., the lysis of infected phytoplankton, producing virus particles on average ().
- A1 .
- A2 .
3. Mathematical Results
3.1. The Ecosystem Without Free Viruses
- 1.
- There are no periodic orbits nor polycycles inside .
- 2.
- If (Assumptions A1–A2), then the equilibrium point exists and it is globally asymptotically stable, and is a saddle point.
- 3.
- If , then the equilibrium point is the unique equilibrium point of system (4), and it is globally asymptotically stable.
- Let be a solution of (4) inside . By using the change in variables and , we obtain the following equations:The divergence of system (4) is given by
- For (Assumptions A1–A2), exists and the Jacobian matrix at is given bySince trace and det , both eigenvalues have a negative real part, and the equilibrium point is therefore a locally asymptotically stable node. The Jacobian matrix evaluated at is therefore given byadmits two eigenvalues that are given byTherefore, the equilibrium is a saddle point with a stable manifold given by . Let and . By applying the Poincaré–Bendixon Theorem [40], the steady state is found to be globally asymptotically stable.
- If , then is the only steady state and is locally stable. Since any omega limit set is contained in the two-dimensional compact invariant set , and lies on the boundary of , is globally asymptotically stable according to the Poincaré-Bendixson Theorem.
3.2. The Complete System
- If , then we obtain a pure epidemic model with a steady state satisfyingThen, , and with . Therefore, we obtain two equilibrium points, and .The development of new infections and changes in the status of infected individuals are described by two equations in our example (Equations (2) and (4) in system (1)). We define the matrix F to represent the rate at which new infections develop in these two equations and V to represent the rate at which individuals are transferred into and out of these compartments by all other mechanisms in order to create the next generation matrix. Next, the non-singular matrix V and the nonnegative matrix F are provided by and . Therefore, . The spectral radius of the next generation matrix FV−1, which can be represented as follows, is the basic reproduction number of (1).Note that . Therefore, exists only if .The Jacobian matrix at is given byadmits , , according to Assumptions A1 and A2, and . Therefore, is an unstable equilibrium point.The Jacobian matrix at is given byThe characteristic polynomial is given byNote thatTherefore, if , then exists, and its Jacobian, , admits three negative eigenvalues and one positive eigenvalue, and thus is an unstable equilibrium point.
- If , then we haveNote that in a population where all phytoplankton are vulnerable to infection in the presence of zooplankton, the basic reproduction number, or , is the anticipated number of instances directly caused by one infected phytoplankton.Let be the Jacobian matrix at .The characteristic polynomial of is given byNote that . Therefore, if , then exists, and its Jacobian, , admits three negative eigenvalues (according to Assumptions A1–A2) and one positive eigenvalue, and thus is an unstable equilibrium point. is a stable equilibrium point only if .Let us discuss the existence and stability of the endemic equilibrium point . Note that means that . Furthermore, we have and . Therefore, if , then , and thus exists. Let be the Jacobian matrix at , which is given byThe characteristic polynomial of is given by, , and . In order to use the Routh–Hurwitz criteria, one must see that and deduce that because . The calculation of the two other conditions was too large, so we used Maple 12 software to verify that and , and thus we obtained the local stability of the equilibrium point once it exists ().
3.3. Reduction to 3D
3.4. The Periodic Orbits on the Faces
- It is easy to see that the axes and are invariant. By applying the following change in the notations and for , one obtains the following model:
- It is easy to see that the axes and are invariant. By applying the following change in the notations and for , one obtains the following model:
3.5. Persistence
- The omega limit set of , represented as , is assumed to include . It should be noted that is a saddle point with a dimension of one stable manifold, , that is limited to the -axis. In this case, is not the whole omega limit set . There is a point, , in , according to the Butler–McGehee lemma [40]. Note that the -axis is unbounded, but that is the -axis. The omega limit set of any orbit of the system (11) should be bounded since all of its orbits are bounded (within the limited set ). The existence of is contradicted by this. Consequently, should be confirmed.
- is assumed. Likewise, should not be the complete omega limit set ; hence, there is a point, , inside . Once is two-dimensional and completely contained in the face, this point should be inside the face. should contain the whole orbit via , just like in the cases of and . In Section 3.4, we demonstrated that there are no possible periodic orbits inside the face. The orbit becomes unbounded once , which runs counter to the assertion that is inside .
- is assumed. Given that is a saddle point, its stable manifold is limited to the plane and has two dimensions. Consequently, is not the whole omega limit set . Accordingly, a point, , exists inside according to the Butler–McGehee lemma [40]. Since lies entirely in the plane, and since the entire orbit through is in , this orbit is thus unbounded, which contradicts the fact that is inside .
3.6. Uniform Persistence of System (7)
- The dynamics are weakly persistent;
- The dynamics are dissipative;
- The restriction of to is isolated;
- The restriction of to is acyclic.
3.7. Uniform Persistence of System (1)
4. Sensitivity Analysis
5. Optimal Control Strategy
6. Numerical Investigations
6.1. Direct Problem
6.2. Optimal Control Problem
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. A Suitable Numerical Scheme for Resolving the Control Problem
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Variable | Description | Units |
---|---|---|
Concentration of susceptible phytoplankton | g phytoplankton/L | |
Concentration of infected phytoplankton | g phytoplankton/L | |
Concentration of zooplankton | g zooplankton/L | |
Concentration of free viruses | g viruses/L | |
Parameter | Description | Units |
D | Dilution rate | h−1 |
Consumption rate of susceptible phytoplankton | h−1 | |
Consumption rate of infected phytoplankton | h−1 | |
Susceptible phytoplankton input concentration | g phytoplankton/L | |
Y | Phytoplankton-to-zooplankton yield coefficient | g zooplankton/g phytoplankton |
Saturated incidence rate | new infections/h | |
Virus replication factor | g infected phytoplankton/g virus |
Parameter | Y | D | |||||
---|---|---|---|---|---|---|---|
Value | 8 |
Parameter, p | Sensitivity Index, | Value |
---|---|---|
0.5 | ||
0.5 | ||
D | −0.0472 | |
−0.217 | ||
−0.4528 | ||
−0.6712 |
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Almuallem, N.A.; El Hajji, M. How Can Viruses Affect the Growth of Zooplankton on Phytoplankton in a Chemostat? Mathematics 2025, 13, 1192. https://doi.org/10.3390/math13071192
Almuallem NA, El Hajji M. How Can Viruses Affect the Growth of Zooplankton on Phytoplankton in a Chemostat? Mathematics. 2025; 13(7):1192. https://doi.org/10.3390/math13071192
Chicago/Turabian StyleAlmuallem, Nada A., and Miled El Hajji. 2025. "How Can Viruses Affect the Growth of Zooplankton on Phytoplankton in a Chemostat?" Mathematics 13, no. 7: 1192. https://doi.org/10.3390/math13071192
APA StyleAlmuallem, N. A., & El Hajji, M. (2025). How Can Viruses Affect the Growth of Zooplankton on Phytoplankton in a Chemostat? Mathematics, 13(7), 1192. https://doi.org/10.3390/math13071192