Dynamics of the oxygen, carbon dioxide, and water interaction across the insect spiracle. (English) Zbl 1470.35370

Simelane, S. M.; Abelman, S.; Duncan, F. D.

Abstr. Appl. Anal. 2014, Article ID 157573, 11 p. (2014).

Summary: This paper explores the dynamics of respiratory gases interactions which are accompanied by the loss of water through an insect’s spiracle. Here we investigate and analyze this interaction by deriving a system of ordinary differential equations for oxygen, carbon dioxide, and water vapor. The analysis is carried out in continuous time. The purpose of the research is to determine bounds for the gas volumes and to discuss the complexity and stability of the equilibria. Numerical simulations also demonstrate the dynamics of our model utilizing the new conditions for stability and instability.

MSC:

35Q92	PDEs in connection with biology, chemistry and other natural sciences
92C30	Physiology (general)

References:

[1]	Alligood, T.; Sauer, T. D.; Yorke, J. A., Chaos: An Introduction to Dynamical Systems (1996), New York, NY, USA: Springer, New York, NY, USA · Zbl 0867.58043
[2]	Cosner, C.; Deangelis, D. L.; Ault, J. S.; Olson, D. B., Effects of spatial grouping on the functional response of predators, Theoretical Population Biology, 56, 1, 65-75 (1999) · Zbl 0928.92031
[3]	Edelstein-Keshet, L., Mathematical Models in Biology (1977), McGraw Hill · Zbl 0674.92001
[4]	Feng, W.; Lu, X.; Liu, W., Comparison and numerical solutions for diffusive models of resource and sexual competition, Nonlinear Analysis: Theory, Methods & Applications, 30, 2765-2774 (1997) · Zbl 0891.35057
[5]	Jansen, V. A., Theoretical aspects of metapopulation dynamics [Ph.D. thesis] (1994), Leiden, The Netherlands: Leiden University, Leiden, The Netherlands
[6]	Månsson, L.; Lundberg, P., An analysis of the analysis of herbivore population dynamics, Oikos, 113, 2, 217-225 (2006)
[7]	Kestler, P.; Hoffmann, K. H., Respiration and respiratory water loss, Environmental Physiology and Biochemistry of Insects, 137-183 (1985), New York, NY, USA: Springer, New York, NY, USA
[8]	Lighton, J. R., Discontinuous ventilation in terrestrial insects, Physiological Zoology, 67, 142-162 (1994)
[9]	Snyder, G. K.; Sheafor, B.; Scholnick, D.; Farrelly, C., Gas exchange in the insect tracheal system, Journal of Theoretical Biology, 172, 3, 199-207 (1995)
[10]	Levy, R. I.; Schneiderman, H. A., Discontinuous respiration in insects-IV. Changes in intratracheal pressure during the respiratory cycle of silkworm pupae, Journal of Insect Physiology, 12, 4, 465-492 (1966)
[11]	Mellany, K., The site of loss of water from insects, Proceedings of The Royal Society of London, 166, 139-149 (1934)
[12]	Welch, W. R.; Tracy, C. R., Respiratory water loss: a predictive model, Journal of Theoretical Biology, 65, 2, 253-265 (1977)
[13]	Woods, H. A.; Smith, J. N., Universal model for water costs of gas exchange by animals and plants, Proceedings of the National Academy of Sciences of the United States of America, 107, 18, 8469-8474 (2010)
[14]	Pickard, W. F., Transition regime diffusion and the structure of the insect tracheolar system, Journal of Insect Physiology, 20, 6, 947-956 (1974)
[15]	Miller, P. L.; Bell, H. J.; Adiyodi, K. G., Respiration, The American Cockroach, 87-116 (1982), London, UK: Chapman & Hall, London, UK
[16]	Gibbs, A. G.; Johnson, R. A., The role of discontinuous gas exchange in insects: the chthonic hypothesis does not hold water, The Journal of Experimental Biology, 207, 20, 3477-3482 (2004)
[17]	Duncan, F. D.; Byrne, M. J., Respiratory airflow in a wingless dung beetle, The Journal of Experimental Biology, 205, 16, 2489-2497 (2002)
[18]	Hetz, S. K.; Bradley, T. J., Insects breathe discontinuously to avoid oxygen toxicity, Nature, 433, 7025, 516-519 (2005)
[19]	Marais, E.; Klok, C. J.; Terblanche, J. S.; Chown, S. L., Insect gas exchange patterns: a phylogenetic perspective, The Journal of Experimental Biology, 208, 23, 4495-4507 (2005)
[20]	Chown, S. L.; Gibbs, A. G.; Hetz, S. K.; Klok, C. J.; Lighton, J. R. B.; Marais, E., Discontinuous gas exchange in insects: a clarification of hypotheses and approaches, Physiological and Biochemical Zoology, 79, 2, 333-343 (2006)
[21]	Wobschall, A.; Hetz, S. K., Oxygen uptake by convection and diffusion in diapausing moth pupae ( Attacus atlas), International Congress Series, 1275, 157-164 (2004)
[22]	Levy, R. I.; Schneiderman, H. A., An experimental solution to the paradox of discontinuous respiration in insects, Nature, 182, 4634, 491-493 (1958)
[23]	Lehmann, F. O., Matching spiracle opening to metabolic need during flight in drosophila, Science, 294, 5548, 1926-1929 (2001)
[24]	Bugboy52.40, Citing Website, Wikipedia
[25]	Arditi, R.; Ginzburg, L. R., Coupling in predator-prey dynamics: ratio-dependence, Journal of Theoretical Biology, 139, 3, 311-326 (1989)
[26]	Hanski, I., The functional response of predators: worries about scale, Trends in Ecology & Evolution, 6, 5, 141-142 (1991)
[27]	Ginzburg, L. R.; Akcakaya, H. R., Consequences of ratio-dependent predation for steady-state properties of ecosystems, Ecology, 73, 5, 1536-1543 (1992)
[28]	Abrams, P. A., The fallacies of ratio-dependent predation, Ecology, 75, 6, 1842-1850 (1994)
[29]	Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 73, 5, 1530-1535 (1992)
[30]	Akcakaya, H. R.; Arditi, R.; Ginzburg, L. R., Ratio-dependent predation: an abstraction that works, Ecology, 76, 3, 995-1004 (1995)
[31]	Abrams, P. A.; Ginzburg, L. R., The nature of predation: prey dependent, ratio dependent or neither?, Trends in Ecology and Evolution, 15, 8, 337-341 (2000)
[32]	Feng, W.; Lu, X.; Donovan, J., Population dynamics in a model for territory acquisition, Discrete and Continuous Dynamical Systems, 156-165 (2001) · Zbl 1301.34014
[33]	Fernandez, X. E. B., A naturalized approach to the autonomy of cognitive agents [Ph.D. thesis] (2008), Leioa, Spain: University of the Basque Country, Leioa, Spain