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Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control
Bifurcation analysis in a delayed toxic-phytoplankton and zooplankton ecosystem with Monod-Haldane functional response
1. | School of Liberal Arts and Sciences, North China Institute of Aerospace Engineering, Langfang, 065000, China |
2. | 283 Company, Second Research Institute of the CASIC, Beijing, 100089, China |
3. | Aerospace Science and Technology, North China Institute of Aerospace Engineering, Langfang, 065000, China |
We structure a phytoplankton zooplankton interaction system by incorporating (i) Monod-Haldane type functional response function; (ii) two delays accounting, respectively, for the gestation delay $ \tau $ of the zooplankton and the time $ \tau_1 $ required for the maturity of TPP. Firstly, we give the existence of equilibrium and property of solutions. The global convergence to the boundary equilibrium is also derived under a certain criterion. Secondly, in the case without the maturity delay $ \tau_1 $, the gestation delay $ \tau $ may lead to stability switches of the positive equilibrium. Then fixed $ \tau $ in stable interval, the effect of $ \tau_1 $ is investigated and find $ \tau_1 $ can also cause the oscillation of system. Specially, when $ \tau = \tau_1 $, under certain conditions, the periodic solution will exist with the wide range as delay away from critical value. To deal with the local stability of the positive equilibrium under a general case with all delays being positive, we use the crossing curve methods, it can obtain the stable changes of positive equilibrium in $ (\tau, \tau_1) $ plane. When choosing $ \tau $ in the unstable interval, the system still can occur Hopf bifurcation, which extends the crossing curve methods to the system exponentially decayed delay-dependent coefficients. Some numerical simulations are given to indicate the correction of the theoretical analyses.
References:
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Q. An, E. Beretta, Y. Kuang, C. Wang and H. Wang,
Geometric stability switch criteria in delay differential equations with two delays and delay dependent parameters, J. Differential Equations, 266 (2019), 7073-7100.
doi: 10.1016/j.jde.2018.11.025. |
[2] |
M. Banerjee and E. Venturino,
A phytoplankton-toxic phytoplankton-zooplankton model, Ecol. Complex., 8 (2011), 239-248.
doi: 10.1016/j.ecocom.2011.04.001. |
[3] |
E. Beretta and Y. Kuang,
Geometric, stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.
doi: 10.1137/S0036141000376086. |
[4] |
P. Bi and S. Ruan,
Bifurcations in delay differential equations and applications to tumor and immune system interaction models, SIAM J. Applied Dynamical Systems, 12 (2013), 1847-1888.
doi: 10.1137/120887898. |
[5] |
J. Chattopadhyay, R. Sarkar and S. Mandal,
Toxin producing plankton may act as a biological control for planktonic blooms: A field study and mathematical modelling, J. Theor. Biol., 215 (2002), 333-344.
doi: 10.1006/jtbi.2001.2510. |
[6] |
J. Chattopadhyay, R. Sarkar and AE Abdllaoui,
A delay differential equation model on harmful algal blooms in the presence of toxic substances, IMA J. Appl. Math., 19 (2002), 137-161.
doi: 10.1093/imammb/19.2.137. |
[7] |
Y. Ding, W. Jiang and P. Yu, Double Hopf bifurcation in delayed vander pol-duffing equation, Internat. J. Bifur. Chaos, 23 (2013), 1350014, 15 pages.
doi: 10.1142/S0218127413500144. |
[8] |
K. Gu, S. Niculescu and J. Chen,
On stability crossing curves for general systems with two delays, J. Math. Anal. Appl., 311 (2005), 231-253.
doi: 10.1016/j.jmaa.2005.02.034. |
[9] |
R. Han and B. Dai, Cross-diffusion induced Turing instability and amplitude equation for a toxic-phytoplankton-zooplankton model with nonmonotonic functional response, Internat. J. Bifur. Chaos, 27 (2017), 1750088, 24 pages.
doi: 10.1142/S0218127417500882. |
[10] |
J. Hale and S. Lunel, Introduction to functional differential Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[11] |
Z. Jiang and T. Zhang,
Dynamical analysis of a reaction-diffusion phytoplankton-zooplankton system with
delay, Chaos, Solitons Fractals, 104 (2017), 693-704.
doi: 10.1016/j.chaos.2017.09.030. |
[12] |
Z. Jiang, W. Zhang, J. Zhang and T. Zhang, Dynamical analysis of a phytoplankton-zooplankton system with harvesting term and Holling III functional response, Internat. J. Bifur. Chaos., 28 (2018), 1850162, 23 pages.
doi: 10.1142/S0218127418501626. |
[13] |
Z. Jiang, J. Dai and T. Zhang, Bifurcation analysis of phytoplankton and zooplanktoninteraction system with two delays, Internat. J. Bifur. Chaos, 30 (2020), 2050039, 21 pages.
doi: 10.1142/S021812742050039X. |
[14] |
H. Jiang and Y. Song,
Normal forms of non-resonance and weak resonance double Hopf bifurcation in the retarded functional differential equations and
applications, Appl. Math. Comput., 266 (2015), 1102-1126.
doi: 10.1016/j.amc.2015.06.015. |
[15] |
Z. Jiang and L. Wang, Global Hopf bifurcation for a predator-prey system with three delays, Internat. J. Bifur. Chaos, 27 (2017), 1750108, 15 pages.
doi: 10.1142/S0218127417501085. |
[16] |
Z. Jiang and Y. Guo, Hopf bifurcation and stability crossing curvein a planktonic resource-consumer system with double delays, Internat. J. Bifur. Chaos, 30 (2020), 2050190, 20 pages.
doi: 10.1142/S0218127420501904. |
[17] |
S. Ma, Q. Lu and Z. Feng,
Double Hopf bifurcation for van der pol-duffing oscillator with parametric delay feedback control, J. Math. Anal. Appl., 338 (2008), 993-1007.
doi: 10.1016/j.jmaa.2007.05.072. |
[18] |
R. Pal, D. Basu and M. Banerjee,
Modelling of phytoplankton allelopathy with
Monod-Haldanetype functional response–A mathematical study, Biosystems, 95 (2009), 243-253.
doi: 10.1016/j.biosystems.2008.11.002. |
[19] |
Y. Qu, J. Wei and S. Ruan,
Stability and bifurcation analysis in hematopoietic stem cell
dynamics with multiple delays, Physica D., 239 (2010), 2011-2024.
doi: 10.1016/j.physd.2010.07.013. |
[20] |
S. Roy, S. Bhattacharya, P. Das and J. Chattopadhyay,
Interaction among non-toxic phytoplankton, toxic phytoplankton and zooplankton:
Inferences from field observations, J. Biol. Phys., 33 (2007), 1-17.
doi: 10.1007/s10867-007-9038-z. |
[21] |
J. Wu,
Symmetric functional differential equations and neural networks with
memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.
doi: 10.1090/S0002-9947-98-02083-2. |
show all references
References:
[1] |
Q. An, E. Beretta, Y. Kuang, C. Wang and H. Wang,
Geometric stability switch criteria in delay differential equations with two delays and delay dependent parameters, J. Differential Equations, 266 (2019), 7073-7100.
doi: 10.1016/j.jde.2018.11.025. |
[2] |
M. Banerjee and E. Venturino,
A phytoplankton-toxic phytoplankton-zooplankton model, Ecol. Complex., 8 (2011), 239-248.
doi: 10.1016/j.ecocom.2011.04.001. |
[3] |
E. Beretta and Y. Kuang,
Geometric, stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.
doi: 10.1137/S0036141000376086. |
[4] |
P. Bi and S. Ruan,
Bifurcations in delay differential equations and applications to tumor and immune system interaction models, SIAM J. Applied Dynamical Systems, 12 (2013), 1847-1888.
doi: 10.1137/120887898. |
[5] |
J. Chattopadhyay, R. Sarkar and S. Mandal,
Toxin producing plankton may act as a biological control for planktonic blooms: A field study and mathematical modelling, J. Theor. Biol., 215 (2002), 333-344.
doi: 10.1006/jtbi.2001.2510. |
[6] |
J. Chattopadhyay, R. Sarkar and AE Abdllaoui,
A delay differential equation model on harmful algal blooms in the presence of toxic substances, IMA J. Appl. Math., 19 (2002), 137-161.
doi: 10.1093/imammb/19.2.137. |
[7] |
Y. Ding, W. Jiang and P. Yu, Double Hopf bifurcation in delayed vander pol-duffing equation, Internat. J. Bifur. Chaos, 23 (2013), 1350014, 15 pages.
doi: 10.1142/S0218127413500144. |
[8] |
K. Gu, S. Niculescu and J. Chen,
On stability crossing curves for general systems with two delays, J. Math. Anal. Appl., 311 (2005), 231-253.
doi: 10.1016/j.jmaa.2005.02.034. |
[9] |
R. Han and B. Dai, Cross-diffusion induced Turing instability and amplitude equation for a toxic-phytoplankton-zooplankton model with nonmonotonic functional response, Internat. J. Bifur. Chaos, 27 (2017), 1750088, 24 pages.
doi: 10.1142/S0218127417500882. |
[10] |
J. Hale and S. Lunel, Introduction to functional differential Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[11] |
Z. Jiang and T. Zhang,
Dynamical analysis of a reaction-diffusion phytoplankton-zooplankton system with
delay, Chaos, Solitons Fractals, 104 (2017), 693-704.
doi: 10.1016/j.chaos.2017.09.030. |
[12] |
Z. Jiang, W. Zhang, J. Zhang and T. Zhang, Dynamical analysis of a phytoplankton-zooplankton system with harvesting term and Holling III functional response, Internat. J. Bifur. Chaos., 28 (2018), 1850162, 23 pages.
doi: 10.1142/S0218127418501626. |
[13] |
Z. Jiang, J. Dai and T. Zhang, Bifurcation analysis of phytoplankton and zooplanktoninteraction system with two delays, Internat. J. Bifur. Chaos, 30 (2020), 2050039, 21 pages.
doi: 10.1142/S021812742050039X. |
[14] |
H. Jiang and Y. Song,
Normal forms of non-resonance and weak resonance double Hopf bifurcation in the retarded functional differential equations and
applications, Appl. Math. Comput., 266 (2015), 1102-1126.
doi: 10.1016/j.amc.2015.06.015. |
[15] |
Z. Jiang and L. Wang, Global Hopf bifurcation for a predator-prey system with three delays, Internat. J. Bifur. Chaos, 27 (2017), 1750108, 15 pages.
doi: 10.1142/S0218127417501085. |
[16] |
Z. Jiang and Y. Guo, Hopf bifurcation and stability crossing curvein a planktonic resource-consumer system with double delays, Internat. J. Bifur. Chaos, 30 (2020), 2050190, 20 pages.
doi: 10.1142/S0218127420501904. |
[17] |
S. Ma, Q. Lu and Z. Feng,
Double Hopf bifurcation for van der pol-duffing oscillator with parametric delay feedback control, J. Math. Anal. Appl., 338 (2008), 993-1007.
doi: 10.1016/j.jmaa.2007.05.072. |
[18] |
R. Pal, D. Basu and M. Banerjee,
Modelling of phytoplankton allelopathy with
Monod-Haldanetype functional response–A mathematical study, Biosystems, 95 (2009), 243-253.
doi: 10.1016/j.biosystems.2008.11.002. |
[19] |
Y. Qu, J. Wei and S. Ruan,
Stability and bifurcation analysis in hematopoietic stem cell
dynamics with multiple delays, Physica D., 239 (2010), 2011-2024.
doi: 10.1016/j.physd.2010.07.013. |
[20] |
S. Roy, S. Bhattacharya, P. Das and J. Chattopadhyay,
Interaction among non-toxic phytoplankton, toxic phytoplankton and zooplankton:
Inferences from field observations, J. Biol. Phys., 33 (2007), 1-17.
doi: 10.1007/s10867-007-9038-z. |
[21] |
J. Wu,
Symmetric functional differential equations and neural networks with
memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.
doi: 10.1090/S0002-9947-98-02083-2. |






Symbol | Parameter Definition | Unit |
Intrinsic growth rate of TPP | day |
|
Environmental carrying capacity | ||
Grazing efficiency of zooplankton | day |
|
Growth efficiency of zooplankton | day |
|
Natural death rate of zooplankton | day |
|
Half-saturation constant | ||
Toxin-producing rate of TPP | ||
Gestation delay of zooplankton | day |
|
Delay required for the maturity of TPP | day |
Symbol | Parameter Definition | Unit |
Intrinsic growth rate of TPP | day |
|
Environmental carrying capacity | ||
Grazing efficiency of zooplankton | day |
|
Growth efficiency of zooplankton | day |
|
Natural death rate of zooplankton | day |
|
Half-saturation constant | ||
Toxin-producing rate of TPP | ||
Gestation delay of zooplankton | day |
|
Delay required for the maturity of TPP | day |
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