March  2020, 25(3): 877-901. doi: 10.3934/dcdsb.2019194

Global dynamics of a reaction-diffusion system with intraguild predation and internal storage

1. 

School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China

2. 

Department of Mathematics and National Center of Theoretical Science, National Tsing-Hua University, Hsinchu 300, Taiwan

3. 

Department of Natural Science in the Center for General Education, Chang Gung University, Guishan, Taoyuan 333, Taiwan

4. 

Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung Branch, Keelung 204, Taiwan

* Corresponding author: fbwang@mail.cgu.edu.tw, fbwang0229@gmail.com

Received  November 2018 Revised  March 2019 Published  March 2020 Early access  September 2019

Fund Project: The first author is supported by NSF of China (11671243, 11572180), the Fundamental Research Funds for the Central Universities (GK201701001).

This paper presents a reaction-diffusion system modeling interactions of the intraguild predator and prey in an unstirred chemostat, in which the predator can also compete with its prey for one single nutrient resource that can be stored within individuals. Under suitable conditions, we first show that there are at least three steady-state solutions for the full system, a trivial steady-state solution with neither species present, and two semitrivial steady-state solutions with just one of the species. Then we establish that coexistence of the intraguild predator and prey can occur if both of the semitrivial steady-state solutions are invasible by the missing species. Comparing with the system without predation, our numerical simulations show that the introduction of predation in an ecosystem can enhance the coexistence of species. Our mathematical arguments also work for the linear food chain model (top-down predation), in which the top-down predator only feeds on the prey but does not compete for nutrient resource with the prey. In our numerical studies, we also do a comparison of intraguild predation and top-down predation.

Citation: Hua Nie, Sze-Bi Hsu, Feng-Bin Wang. Global dynamics of a reaction-diffusion system with intraguild predation and internal storage. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 877-901. doi: 10.3934/dcdsb.2019194
References:
[1]

C. J. Bampfylde and M. A. Lewis, Biological control through intraguild predation: Case studies in pest control, invasive species and range expansion, Bull. Math. Biol., 69 (2007), 1031-1066.  doi: 10.1007/s11538-006-9158-9.

[2]

S. Diehl, Relative consumer sizes and the strengths of direct and indirect interactions in omnivorous feeding relationships, Oikos, 68 (1993), 151-157.  doi: 10.2307/3545321.

[3]

S. Diehl, Direct and indirect effects of omnivory in a littoral lake community, Ecology, 76 (1995), 1727-1740.  doi: 10.2307/1940706.

[4]

S. Diehl and M. Feissel, Effects of enrichment on threelevel food chains with omnivory, Am. Nat., 155 (2000), 200-218.  doi: 10.1086/303319.

[5]

S. Diehl and M. Feissel, Intraguild prey suffer from enrichment of their resources: a microcosm experiment with ciliates, Ecology, 82 (2001), 2977-2983.  doi: 10.2307/2679828.

[6]

M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272.  doi: 10.1111/j.1529-8817.1973.tb04092.x.

[7]

J. P. Grover, Resource Competition, Chapman and Hall, London, 1997. doi: 10.1007/978-1-4615-6397-6.

[8]

J. P. Grover, Resource competition in a variable environment: phytoplankton growing according to the variable-internal-stores model, Am. Nat., 138 (1991), 811-835. 

[9]

J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, Am. Nat., 178 (2011), 124-148.  doi: 10.1086/662163.

[10]

J. P. GroverS. B. Hsu and F.-B. Wang, Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation, J. Math. Biol., 64 (2012), 713-743.  doi: 10.1007/s00285-011-0426-4.

[11]

J. P. Grover and F.-B. Wang, Competition for one nutrient with internal storage and toxin mortality, Math. Biosci., 244 (2013), 82-90.  doi: 10.1016/j.mbs.2013.04.009.

[12]

J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society Providence, RI, 1988.

[13]

W. M. HirschH. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellers for semidynamical systems, J. Dynam. Differential Equations, 13 (2001), 107-131.  doi: 10.1023/A:1009044515567.

[14]

R. D. Holt and G. A. Polis, A theoretical framework for intraguild predation, Am. Nat., 149 (1997), 745-764.  doi: 10.1086/286018.

[15]

S. B. HsuS. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383.  doi: 10.1137/0132030.

[16]

S. B. HsuJ. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Differential Equations, 248 (2010), 2470-2496.  doi: 10.1016/j.jde.2009.12.014.

[17]

S. B. HsuK.-Y. Lam and F. B. Wang, Single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat, J. Math. Biol., 75 (2017), 1775-1825.  doi: 10.1007/s00285-017-1134-5.

[18]

S. B. HsuJ. P. Shi and F. B. Wang, Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3169-3189.  doi: 10.3934/dcdsb.2014.19.3169.

[19]

S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unstirred Chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044.  doi: 10.1137/0153051.

[20]

P. Magal and X. -Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.

[21]

J. Mallet-Paret and R. D. Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theor. Appl., 7 (2010), 103-143.  doi: 10.1007/s11784-010-0010-3.

[22]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.

[23]

L. MeiS. B. Hsu and F.-B. Wang, Growth of single phytoplankton species with internal storage in a water column, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 607-620.  doi: 10.3934/dcdsb.2016.21.607.

[24]

F. M. M. Morel, Kinetics of nutrient uptake and growth in phytoplankton, J. Phycol., 23 (1987), 137-150.  doi: 10.1111/j.1529-8817.1987.tb04436.x.

[25]

H. NieS.-B. Hsu and F.-B. Wang, Steady-state solutions of a reaction-diffusion system arising from intraguild predation and internal storage, J. Differential Equations, 266 (2019), 8459-8491.  doi: 10.1016/j.jde.2018.12.035.

[26] H. NieJ. H. Wu and Z. G. Wang, Dynamics on the Unstirred Chemostat Models, Science Press, Beijing, 2017. 
[27]

G. A. Polis and et al, The ecology and evolution of intraguild predation: Potential competitors that eat each other, Annu. Rew. Ecol. Syst., 20 (1989), 297-330.  doi: 10.1146/annurev.es.20.110189.001501.

[28]

G. A. Polis and R. D. Holt, Intraguild predation: the dynamics of complex trophic interactions, Trends Ecol. Evol., 7 (1992), 151-154.  doi: 10.1146/annurev.es.20.110189.001501.

[29]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.

[30]

J. A. RosenheimH. K. KayaL. E. EhleretJ. J. Marois and B. A. Jaffee, Intraguild predation among biological control agents: Theory and evidence, Biol. Control, 5 (1995), 303-335.  doi: 10.1006/bcon.1995.1038.

[31]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995.

[32]

H. L. Smith and P. Waltman, Competition for a single limiting resouce in continuous culture: the variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.  doi: 10.1137/S0036139993245344.

[33] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge Univ. Press, 1995.  doi: 10.1017/CBO9780511530043.
[34]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[35]

F.-B. WangS.-B. Hsu and Y.-H. Ho, Mathematical analysis on a Droop model with intraguild predation, Taiwanese J. Math., 23 (2019), 351-373.  doi: 10.11650/tjm/181011.

[36]

S. WilkenJ. M. H. VerspagenS. Naus-WiezerE. V. Donk and J. Huisman, Comparison of predator-prey interactions with and without intraguild predation by manipulation of the nitrogen source, Oikos, 123 (2014), 423-432.  doi: 10.1111/j.1600-0706.2013.00736.x.

[37]

J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835.  doi: 10.1016/S0362-546X(98)00250-8.

[38]

J. H. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the un-stirred chemostat, J. Differential Equations, 172 (2001), 300-332.  doi: 10.1006/jdeq.2000.3870.

[39]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-3-319-56433-3.

show all references

References:
[1]

C. J. Bampfylde and M. A. Lewis, Biological control through intraguild predation: Case studies in pest control, invasive species and range expansion, Bull. Math. Biol., 69 (2007), 1031-1066.  doi: 10.1007/s11538-006-9158-9.

[2]

S. Diehl, Relative consumer sizes and the strengths of direct and indirect interactions in omnivorous feeding relationships, Oikos, 68 (1993), 151-157.  doi: 10.2307/3545321.

[3]

S. Diehl, Direct and indirect effects of omnivory in a littoral lake community, Ecology, 76 (1995), 1727-1740.  doi: 10.2307/1940706.

[4]

S. Diehl and M. Feissel, Effects of enrichment on threelevel food chains with omnivory, Am. Nat., 155 (2000), 200-218.  doi: 10.1086/303319.

[5]

S. Diehl and M. Feissel, Intraguild prey suffer from enrichment of their resources: a microcosm experiment with ciliates, Ecology, 82 (2001), 2977-2983.  doi: 10.2307/2679828.

[6]

M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272.  doi: 10.1111/j.1529-8817.1973.tb04092.x.

[7]

J. P. Grover, Resource Competition, Chapman and Hall, London, 1997. doi: 10.1007/978-1-4615-6397-6.

[8]

J. P. Grover, Resource competition in a variable environment: phytoplankton growing according to the variable-internal-stores model, Am. Nat., 138 (1991), 811-835. 

[9]

J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, Am. Nat., 178 (2011), 124-148.  doi: 10.1086/662163.

[10]

J. P. GroverS. B. Hsu and F.-B. Wang, Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation, J. Math. Biol., 64 (2012), 713-743.  doi: 10.1007/s00285-011-0426-4.

[11]

J. P. Grover and F.-B. Wang, Competition for one nutrient with internal storage and toxin mortality, Math. Biosci., 244 (2013), 82-90.  doi: 10.1016/j.mbs.2013.04.009.

[12]

J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society Providence, RI, 1988.

[13]

W. M. HirschH. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellers for semidynamical systems, J. Dynam. Differential Equations, 13 (2001), 107-131.  doi: 10.1023/A:1009044515567.

[14]

R. D. Holt and G. A. Polis, A theoretical framework for intraguild predation, Am. Nat., 149 (1997), 745-764.  doi: 10.1086/286018.

[15]

S. B. HsuS. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383.  doi: 10.1137/0132030.

[16]

S. B. HsuJ. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Differential Equations, 248 (2010), 2470-2496.  doi: 10.1016/j.jde.2009.12.014.

[17]

S. B. HsuK.-Y. Lam and F. B. Wang, Single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat, J. Math. Biol., 75 (2017), 1775-1825.  doi: 10.1007/s00285-017-1134-5.

[18]

S. B. HsuJ. P. Shi and F. B. Wang, Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3169-3189.  doi: 10.3934/dcdsb.2014.19.3169.

[19]

S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unstirred Chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044.  doi: 10.1137/0153051.

[20]

P. Magal and X. -Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.

[21]

J. Mallet-Paret and R. D. Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theor. Appl., 7 (2010), 103-143.  doi: 10.1007/s11784-010-0010-3.

[22]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.

[23]

L. MeiS. B. Hsu and F.-B. Wang, Growth of single phytoplankton species with internal storage in a water column, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 607-620.  doi: 10.3934/dcdsb.2016.21.607.

[24]

F. M. M. Morel, Kinetics of nutrient uptake and growth in phytoplankton, J. Phycol., 23 (1987), 137-150.  doi: 10.1111/j.1529-8817.1987.tb04436.x.

[25]

H. NieS.-B. Hsu and F.-B. Wang, Steady-state solutions of a reaction-diffusion system arising from intraguild predation and internal storage, J. Differential Equations, 266 (2019), 8459-8491.  doi: 10.1016/j.jde.2018.12.035.

[26] H. NieJ. H. Wu and Z. G. Wang, Dynamics on the Unstirred Chemostat Models, Science Press, Beijing, 2017. 
[27]

G. A. Polis and et al, The ecology and evolution of intraguild predation: Potential competitors that eat each other, Annu. Rew. Ecol. Syst., 20 (1989), 297-330.  doi: 10.1146/annurev.es.20.110189.001501.

[28]

G. A. Polis and R. D. Holt, Intraguild predation: the dynamics of complex trophic interactions, Trends Ecol. Evol., 7 (1992), 151-154.  doi: 10.1146/annurev.es.20.110189.001501.

[29]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.

[30]

J. A. RosenheimH. K. KayaL. E. EhleretJ. J. Marois and B. A. Jaffee, Intraguild predation among biological control agents: Theory and evidence, Biol. Control, 5 (1995), 303-335.  doi: 10.1006/bcon.1995.1038.

[31]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995.

[32]

H. L. Smith and P. Waltman, Competition for a single limiting resouce in continuous culture: the variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.  doi: 10.1137/S0036139993245344.

[33] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge Univ. Press, 1995.  doi: 10.1017/CBO9780511530043.
[34]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[35]

F.-B. WangS.-B. Hsu and Y.-H. Ho, Mathematical analysis on a Droop model with intraguild predation, Taiwanese J. Math., 23 (2019), 351-373.  doi: 10.11650/tjm/181011.

[36]

S. WilkenJ. M. H. VerspagenS. Naus-WiezerE. V. Donk and J. Huisman, Comparison of predator-prey interactions with and without intraguild predation by manipulation of the nitrogen source, Oikos, 123 (2014), 423-432.  doi: 10.1111/j.1600-0706.2013.00736.x.

[37]

J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835.  doi: 10.1016/S0362-546X(98)00250-8.

[38]

J. H. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the un-stirred chemostat, J. Differential Equations, 172 (2001), 300-332.  doi: 10.1006/jdeq.2000.3870.

[39]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-3-319-56433-3.

Figure 1.  The effects of the nutrient supply concentration $R^{(0)} $: (A, C) Intraguild predation with $a_{\max,2} = 24.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $, and (B, D) top-down predation with $a_{\max,2} = 0\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $. $R^{(0)} = 1.5\times10^{-5}\, \mathrm{mol}\,\mathrm{l}^{-1}$ in (A, B), and $R^{(0)} = 2.5\times10^{-5}\, \mathrm{mol}\,\mathrm{l}^{-1}$ in (C, D)
Figure 2.  Bifurcation diagrams of positive steady state solutions to (5)-(7) with the bifurcation parameter $R^{(0)}$ ranging from $0.5\times10^{-5}$ to $2.0\times10^{-4}\, \mathrm{mol}\,\mathrm{l}^{-1}.$ (A) Intraguild predation with $a_{\max,2} = 24.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $, and (B) top-down predation with $a_{\max,2} = 0\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $
Figure 3.  The effects of the diffusion rate $d $: (A, C, E) Intraguild predation with $a_{\max,2} = 24.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $, and (B, D, F) top-down predation with $a_{\max,2} = 0\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $. $d = 0.08\, \mathrm{day}^{-1}$ in (A, B), $d = 0.12\, \mathrm{day}^{-1}$ in (C, D), and $d = 0.16\, \mathrm{day}^{-1}$ in (E, F)
Figure 4.  Bifurcation diagrams of positive steady state solutions to (5)-(7) with the bifurcation parameter $g_{\max}$ ranging from $0$ to $120\ \mathrm{cells}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $. (A) Intraguild predation with $a_{\max,2} = 24.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $, and (B) top-down predation with $a_{\max,2} = 0\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $
Table 1.  Common parameters used in intraguild predation and top-down predation
Quantity Value Quantity Value
$\gamma $ $10\, \mathrm{day}^{-1} $ $a_{\max,1} $ $12.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $
$K_1 $ $9.0\times10^{-7}\, \mathrm{mol}\,\mathrm{l}^{-1} $ $K_2 $ $6.5\times10^{-7}\, \mathrm{mol}\,\mathrm{l}^{-1} $
$\mu_{\max,1} $ $0.7\,\mathrm{day}^{-1} $ $\mu_{\max,2} $ $2.2\,\mathrm{day}^{-1} $
$Q_{\min,1} $ $2.6\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1} $ $Q_{\min,2} $ $1.0\times10^{-13}\, \mathrm{mol}\,\mathrm{cell}^{-1} $
$Q_{\max,1} $ $9.5\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1} $ $Q_{\max,2} $ $32.0\times10^{-13}\, \mathrm{mol}\,\mathrm{cell}^{-1} $
$b $ 2.37 $K_0 $ $4.0\times10^{8}\,\mathrm{cells}\,\mathrm{l}^{-1} $
Quantity Value Quantity Value
$\gamma $ $10\, \mathrm{day}^{-1} $ $a_{\max,1} $ $12.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $
$K_1 $ $9.0\times10^{-7}\, \mathrm{mol}\,\mathrm{l}^{-1} $ $K_2 $ $6.5\times10^{-7}\, \mathrm{mol}\,\mathrm{l}^{-1} $
$\mu_{\max,1} $ $0.7\,\mathrm{day}^{-1} $ $\mu_{\max,2} $ $2.2\,\mathrm{day}^{-1} $
$Q_{\min,1} $ $2.6\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1} $ $Q_{\min,2} $ $1.0\times10^{-13}\, \mathrm{mol}\,\mathrm{cell}^{-1} $
$Q_{\max,1} $ $9.5\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1} $ $Q_{\max,2} $ $32.0\times10^{-13}\, \mathrm{mol}\,\mathrm{cell}^{-1} $
$b $ 2.37 $K_0 $ $4.0\times10^{8}\,\mathrm{cells}\,\mathrm{l}^{-1} $
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