A two-species predator-prey model in an environment enriched by a biotic resource

Classical population growth models assume that the environmental carrying capacity is a fixed parameter, which is not often realistic. We propose a modified predator-prey model where the carrying capacity of the environment is dependent on the availability of a biotic resource. In this model both populations are able to consume the resource, thus altering the environment. Stability, bifurcation and numerical analyses are presented to illustrate the system's dynamical behaviour. Bistability occurs in certain parameter regions. This could describe the transition from a beneficial environment to a detrimental one. We examine special cases of the system and show that both permanence and extinction are possible. References J. Vandermeer. Seasonal isochronic forcing of Lotka Volterra equations. Prog. Theor. Phys., 96:13–28, 1996. doi:10.1143/PTP.96.13 S. Ikeda and T. Yokoi. Fish population dynamics under nutrient enrichment–-A case of the East Seto Inland Sea. Ecol. Model., 10:141–165, 1980. doi:10.1016/0304-3800(80)90057-5 S. P. Rogovchenko and Y. V. Rogovchenko. Effect of periodic environmental fluctuations on the Pearl–Verhulst model. Chaos, Solitons, Fractals, 39:1169–1181, 2009. doi:10.1016/j.chaos.2007.11.002 H. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu. A simple model for the total microbial biomass under occlusion of healthy human skin. In Chan, F., Marinova, D. and Anderssen, R.S. (eds) MODSIM2011, 19th International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand., 733–739, 2011. http://www.mssanz.org.au/modsim2011/AA/safuan.pdf P. Meyer and J. H. Ausubel. Carrying capacity: A model with logistically varying limits. Technol. Forecast. Soc., 61:209–214, 1999. doi:10.1016/S0040-1625(99)00022-0 R. Huzimura and T. Matsuyama. A mathematical model with a modified logistic approach for singly peaked population processes. Theor. Popul. Biol., 56:301–306, 1999. doi:10.1006/tpbi.1999.1426 J. H. M. Thornley and J. France. An open-ended logistic-based growth function. Ecol. Model., 184:257–261, 2005. doi:10.1016/j.ecolmodel.2004.10.007 J. H. M. Thornley, J. J. Shepherd and J. France. An open-ended logistic-based growth function: Analytical solutions and the power-law logistic model. Ecol. Model., 204:531–534, 2007. doi:10.1016/j.ecolmodel.2006.12.026 H. M. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu. Coupled logistic carrying capacity model. ANZIAM J, 53:C172–C184, 2012. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/4972 P. H. Leslie and J. C. Gower. The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrika, 47:219–234, 1960. doi:10.1093/biomet/47.3-4.219 B. Basener and D. S. Ross. Booming and crashing populations and Easter Island. SIAM J. Appl. Math., 65:684–701, 2005. doi:10.1137/S0036139903426952 D. Lacitignola and C. Tebaldi. Symmetry breaking effects on equilibria and time dependent regimes in adaptive Lotka–Volterra systems. Int. J. Bifurcat. Chaos, 13:375–392, 2003. doi:10.1142/S0218127403006595 F. Wang and G. Pang. Chaos and Hopf bifurcation of a hybrid ratio-dependent three species food chain. Chaos, Solitons, Fractals, 36:1366–1376, 2008. doi:10.1016/j.chaos.2006.09.005 R. Ball. Understanding critical behaviour through visualization: A walk around the pitchfork. Comput. Phys. Commun., 142:71–75, 2001. doi:10.1016/S0010-4655(01)00322-8 B. Ermentrout. XPP-Aut v. 6.00, 2011. http://www.math.pitt.edu/ bard/xpp/xpp.html R. Malka, E. Shochat and V. R. Kedar. Bistability and bacterial infections. PLOS ONE, 5:1–10, 2010. doi:10.1371/journal.pone.0010010 J. Elf, K. Nilsson, T. Tenson and M. Ehrenberg. Bistable bacterial growth rate in response to antibiotics with low membrane permeability. Phys. Rev. Lett., 97:258104, 2006. doi:10.1103/PhysRevLett.97.258104 D. Dubnau and R. Losick. Bistability in bacteria. Mol. Microbiol., 61:564–572, 2006. doi:10.1111/j.1365-2958.2006.05249.x M. Santillan. Bistable behavior in a model of the lac Operon in Escherichia coli with variable growth rate. Biophys. J., 94:2065–2081, 2008. doi:10.1529/biophysj.107.118026 M. Rosenzweig. Paradox of enrichment: destabilization of exploitation ecosystem in ecological time. Science, 171:385–387, 1971. doi:10.1126/science.171.3969.38

Hierarchical finite elements for stress concentration and stress singularity problems

The p-version finite element method offers a distinct advantage of savings in computational time and effort in comparison with the conventional, h-version finite element method. The h-version uses low order elements and convergence studies are done by successive refinements of the mesh that imply analyzing the problem afresh. In contrast, the p-version uses a fixed discretization of the domain and higher order elements are successively employed during convergence studies. The computational advantage results from the use of hierarchical shape functions, coarse meshes and faster rates of convergence with decreased number of degrees of freedom. Consequently, the use of hierarchical finite elements can be especially advantageous for stress concentration and stress singularity problems that require very refined meshes in the h-version. -- The accuracy of hierarchical finite elements is demonstrated with the use of coarse meshes for beam and T-plate weld joint problems. A 2D enriched hierarchical finite element is developed for stress intensity factor evaluation that embodies the inverse square root stress singularity by including in its formulation the stress intensity factors as additional degrees of freedom. Stress intensity factors for cracked specimens are numerically evaluated quite accurately using very coarse meshes involving fewer degrees of freedom in comparison with conventional analyses. This concept is then extended to 3D crack problems and favourable results are obtained

Understanding the role of moisture in the self-heating process of compost piles

This paper considers the self-heating process which occurs in a compost pile using one-dimensional spatially-dependent models and incorporating terms that account for self-heating due to both biological and oxidation mechanisms. As the moisture content in a compost pile is a crucial factor in its degradation process, we utilise a model which incorporates four mass-balance equations, namely, energy, oxygen, vapour and liquid water concentrations, to investigate the behaviour of compost piles when moisture content is present. Analyses of different initial water contents within a compost pile, different ambient relative humidities and different amounts of water added to the pile by rainstorms are undertaken. We show that the effects of the ambient relative humidity are not significant but that a rainstorm either accelerates or decelerates a compost pile\u27s self-heating process significantly depending on the initial moisture content of the compost materials and the amount of water that is added

An analysis of a standard reactor cascade and a step-feed reactor cascade for biological processes described by monod kinetics

We analyse the steady-state operation of two types of reactor cascade without recycle. The first is a standard reactor cascade in which the feed stream enters into the first reactor. The second is a step-feed reactor cascade in which an equal proportion of the feed stream enters each reactor in the cascade. The reaction is assumed to be a biological process governed by Monod growth kinetics with a decay coefficient for the microorganisms. The steady-states of both models are found for an arbitrary number of reactors and their stability determined as a function of the residence time. We show that in a step-feed reactor cascade the substrate and biomass concentrations leaving the reactor of the cascade are identical to those leaving the first reactor of the cascade. We further show that this result is true for a general specific growth rate of the form μ (S,X). Thus for such processes the non-standard cascade offers no advantage over that of a single reactor. This is surprising because the use of a non-standard cascade has been proposed as a mechanism to improve the biological treatment of wastewater

The passage of food through animal stomachs: a chemical reactor engineering approach

In many circumstances it is useful to know the mean residence time of food substrates within the body following digestion. For instance, such information is crucial to estimate the extent to which dietary components are fermented inside animal stomachs. The mean residence time can be estimated by measuring the rate at which non-absorbable markers, mixed as a supplement into an animals food, are deposited in the animals faeces. The experimental data are analysed with the use of an appropriate mathematical model. We analyse multicompartmental models for the flow of digesta along the gastrointestinal tract of animals. The problem can be treated as a sequence of `tanks\u27 in series. Of interest is the fact that the volume of the tanks is not necessarily constant. For example, following digestion of food, secretion of pancreatic juices may occur; diluting the tracer. Thus the problem can be treated as a series of semi-batch reactors in series. This problem is a good illustration of the application of the methods of chemical reactor engineering to a situation that, at first sight, does not appear to be a chemical engineering problem

Mathematical analysis of the activated aludge process for domestic wastewater treatment

We analyse a model for the activated sludge process occurring in a biological reactor without recycle. The biochemical processes occurring within the reactor are represented by the activated sludge model number 1 (ASM1). In the past the ASM1 model has been investigated via direct integration of the governing equations. This approach is time consuming as parameter regions of interest (in terms of the effluent quality leaving the plant) can only be determined through laborious and repetitive calculations. In this work we use continuation methods to determine the steady-state behaviour of the system. In particular, we determine bifurcation values of the residence time, corresponding to branch points, that are crucial in determining the performance of the plant

A Theoretical Investigation into Phase Change Clothing Benefits for Firefighters under Extreme Conditions

We investigate the thermal performance of protective clothing that has an embedded phase change layer. Heat absorption due to phase change within the material is used to limit the thermal penetration of heat into the material and hence to the firefighter. The distribution of temperature within the fabric and skin during the exposure to an extreme firefighting situation is determined. To determine the protective nature of the clothing, we also include a model of the skin as three layers with differing thermal properties namely the epidermis, dermis and the subcutaneous layer. In our model, we have also incorporated the air gap between the garment and the body. The mathematical model is used to predict the duration of fire exposure during which the garment is able to protect the firefighter from getting first and second degree burns

Performance analysis of the activated sludge model (number 1)

The activated sludge process is widely used to treat both municipal sewage and a variety of industrial wastewaters. We investigate the steady-state behaviour of an activated sludge process. We use the activated sludge model number one, an internationally accepted model, to describe the biochemical, biological, and physical-chemical phenomena that occur inside the bioreactor. The treatment configuration consists of a single aerated reactor attached to a settling unit. Continuation methods are used to determine the steady-states of the model as a function of the hydraulic residence time. From these solutions we construct important operational parameters including the chemical oxygen demand, total suspended solids and total nitrogen. These are determined inside the bioreactor, in the effluent stream and in the wastage stream. We show that there are two critical values of the hydraulic retention time. As the hydraulic retention time is increased through the first critical value heterotrophic biomass become viable. This bifurcation is associated with a substantial decrease in the chemical oxygen demand in the effluent stream and a corresponding increase in both the total suspended solids and total nitrogen in the reactor. Autotrophic biomass become viable as the hydraulic retention time is increased through the second bifurcation point. Associated with this bifurcation there are dramatic changes in the concentration of soluble ammonium nitrogen and soluble nitrate/nitrite inside the reactor; the former being converted to the latter. Of particular practical interest is the value of the hydraulic retention time at which the chemical oxygen demand in the effluent stream is equal to a preset target value. We investigate how this value varies as either the composition of the influent stream or the recycle ratio is varied