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The provision of additional food (AF) sources to an introduced predator has been identified as a mechanism to improve pest control. However, AF models with prey dependent functional responses can cause unbounded growth of the predator \cite{S27}. To avoid such dynamics, an AF model with mutual interference effect has been proposed \cite{S02}. The analysis therein reveals that if the quantity of additional food $\xi > h(\epsilon)$, where $\epsilon$ is the mutual interference parameter, then pest eradication is possible, and this is facilitated via a transcritical bifurcation. We revisit this model and show novel dynamical behaviors. In particular, pest eradication is possible for a tighter range of AF $g(\epsilon) < \xi < f(\epsilon) < h(\epsilon)$, and can also occur via a saddle node bifurcation. We observe bi-stability, as well as local bifurcations of Hopf type. We also prove a global bifurcation, of homoclinic type. This bifurcation in turn is shown to create a non-standard dynamic wherein the pest extinction state becomes an ``almost" global attractor. To the best of our knowledge, this is the first proof of existence of such a dynamical structure in AF models. We discuss our analysis in the context of designing novel bio-control strategies.

The classical chemostat is an intensely investigated model in ecology and bio/chemical engineering, where n-species, say $x_{1}, x_{2}...x_{n}$ compete for a single growth limiting nutrient. Classical theory predicts that depending on model parameters, one species competitively excludes all others. Furthermore, this ''order'' of strongest to weakest is preserved, $x_{1} >> x_{2} >> ...x_{n}$, for say $D_{1} < D_{2} <...D_{n}$, where $D_{i}$ is the net removal of species $x_{i}$. Meaning $x_{1}$ is the strongest or most dominant species and $x_{n}$ is the weakest or least dominant. We propose a modified version of the classical chemostat, exhibiting certain counterintuitive dynamics. Herein we show that if only a certain proportion of the weakest species $x_{n}$'s population is removed at a ''very'' fast density dependent rate, it will in fact be able to competitively exclude all other species, for certain initial conditions. Numerical simulations are carried out to visualize these dynamics in the three species case., Comment: 14 pages, 4 figures

A number of top down bio-control models have been proposed where the predators' efficacy is enhanced via the provision of additional food (AF). These can lead to infinite time \emph{blow-up} of the predator . Predator interference, predator competition, refuge, pest defense and cannibalism have been proposed to achieve pest extinction, while keeping the predator population bounded, but entail \emph{quadratic} or higher order damping terms. Thus an AF model with monotone pest dependent functional response, devoid of these mechanisms, cannot yield the pest extinction state. In the current manuscript we propose a novel patch driven bio-control model, with monotone pest dependent functional response. Here a predator is introduced into a ``patch" such as a prairie strip with AF, and then disperses into a neighboring ``patch" such as a crop field, to target a pest. Assuming only \emph{linear} dispersal in the predator, we show, (i) blow-up in the predator can be prevented in both the prairie strip and the crop field, (ii) the pest extinction state is attainable in the crop field, (iii) the AF model with patch structure can keep pest densities lower than the AF model without patch structure and/or the classical top-down bio-control model, (iv) the AF model with patch structure possesses rich dynamics including chaos. In particular, one observes ``patch specific chaos" - the pest occupying the crop-field can oscillate chaotically, while the pest in the prairie strip oscillates periodically. We discuss these results in light of bio-control strategies that would utilize prairie strips., Comment: 21 figures, 29 pages

In the current manuscript, a first two-patch model with Allee effect and nonlinear dispersal is presented. We study both the ODE case and the PDE case here. In the ODE model, the stability of the equilibrium points and the existence of saddle-node bifurcation are discussed. The phase diagram and bifurcation curve of our model are also given by numerical simulation. Besides, the corresponding linear dispersal case is also presented. We show that when the Allee effect is large, high intensity of linear dispersal is not favorable to the persistence of the species. We further show when the Allee effect is large, nonlinear diffusion is more favorable to the survival of the population than linear diffusion. Moreover, the results of the PDE model extends our findings from discrete patches to continuous patches.

Aphids are damaging insect pests on many crops. Their density can rapidly build up on a host plant to several thousand over one growing season. Occasionally, a competition-driven decline in population early in the season, followed by a build-up later, is observed in the field. Such dynamics cannot be captured via standard models, such as introduced in Kindlmann and Dixon, 2010. In Kindlmann et al., 2007, a logistic non-local population model with variable carrying capacity is proposed to capture these alternative dynamics. The proposed model has a rich dynamical structure and can predict multiple population peaks, as observed in the field. We show that additionally, this model also possesses solutions that can blow-up/explode in finite time. The blow-up is seen to occur for both large and small initial conditions and can occur both early and late in the season. We propose a model extension that, under certain parametric restrictions, has global time-bounded solutions. We discuss the ecological applications of our findings., Comment: 21 pages, 8 figures

This paper is the first to propose an allelopathic phytoplankton competition ODE model influenced by a fear effect based on natural biological phenomena. It is shown that the interplay of this fear effect and the allelopathic term cause rich dynamics in the proposed competition model, such as global stability, transcritical bifurcation, pitchfork bifurcation, and saddle-node bifurcation. We also consider the spatially explicit version of the model and prove analogous results. Numerical simulations verify the feasibility of the theoretical analysis. The results demonstrate that the primary cause of the extinction of non-toxic species is the fear of toxic species compared to toxins. Allelopathy only affects the density of non-toxic species. The discussion provides guidance for the conservation of species and the maintenance of biodiversity., Comment: arXiv admin note: text overlap with arXiv:2303.04919

In the current manuscript, we consider a predator-prey model where the predator is modeled as a generalist using a modified Leslie-Gower scheme, and the prey exhibits group defence via a generalised response. We show that the model could exhibit finite time blow-up, contrary to the current literature (Eur. Phys. J. Plus 137, 28). We also propose a new concept via which the predator population blows up in finite time while the prey population quenches in finite time. The blow-up and quenching times are proved to be one and the same. Our analysis is complemented by numerical findings. This includes a numerical description of the basin of attraction for large data blow-up solutions, as well as several rich bifurcations leading to multiple limit cycles, both in co-dimension one and two. Lastly, we posit a delayed version of the model with globally existing solutions for any initial data.

We consider the effect of finite time extinction mechanisms (FTEM) such as (1) semi-linear harvesting terms, and (2) quasi-linear fast diffusion terms on two species Lokta-Volterra competition models. We show that these mechanisms can alter classical dynamics of competitive exclusion, and weak and strong competition by acting only on a \emph{small} portion of the weaker competitors' population, analogous to small defector populations in game theory \cite{DC23}. In particular, a stronger competitors population, with a few individuals dispersing (``defecting") very quickly, could exhibit bi-stability, as well as competitive exclusion \emph{reversal}. The non-linear harvesting is applied to aphid-soybean crop systems, wherein novel dynamics are observed. Applications to bio-control of invasive pests such as the soybean aphid are discussed., Comment: 30 pages

Population ecology theory is replete with density dependent processes. However trait-mediated or behavioral indirect interactions can both reinforce or oppose density-dependent effects. This paper presents the first two species competitive ODE and PDE systems where an Allee effect, which is a density dependent process and the fear effect, which is non-consumptive and behavioral are both present. The stability of the equilibria is discussed analytically using the qualitative theory of ordinary differential equations. It is found that the Allee effect and the fear effect change the extinction dynamics of the system and the number of positive equilibrium points, but they do not affect the stability of the positive equilibria. We also observe some special dynamics that induce bifurcations in the system by varying the Allee or fear parameter. Interestingly we find that the Allee effect working in conjunction with the fear effect, can bring about several qualitative changes to the dynamical behavior of the system with only the fear effect in place, in regimes of small fear. That is, for small amounts of the fear parameter, it can change a competitive exclusion type situation to a strong competition type situation. It can also change a weak competition type situation to a bi-stability type situation. However for large fear regimes the Allee effect reinforces the dynamics driven by the fear effect. The analysis of the corresponding spatially explicit model is also presented. To this end the comparison principle for parabolic PDE is used. The conclusions of this paper have strong implications for conservation biology, biological control as well as the preservation of biodiversity.

Non-consumptive effects such as fear of depredation, can strongly influence predator-prey dynamics. These effects have not been as well studied in the case of purely competitive systems, despite ecological and social motivations for the same. In this work we consider the classic two species ODE and PDE Lokta-Volterra competition models, where \emph{one} of the competitors is "fearful" of the other. We find that the presence of fear can have several interesting dynamical effects on the classical scenarios of weak and strong competition, and competitive exclusion. Notably, for fear levels in certain regimes, we show bi-stability between interior equilibrium and boundary equilibrium is possible - contrary to the classical strong competition situation where bi-stability is only possible between boundary equilibrium. Furthermore, in the spatially explicit setting, the effects of several spatially heterogeneous fear functions are investigated. In particular, we show that under certain $\mathbb{L}^{1}$ restrictions on the fear function, a weak competition type situation can change to competitive exclusion. Applications of these results to ecological as well as sociopolitical settings are discussed, that connect to the "landscape of fear" (LOF) concept in ecology., Comment: 42 pages, 18 figures

The literature posits that an introduced predator population, is able to drive it's target pest population extinct, if supplemented with high quality additional food of quantity $\xi > \xi_{critical}$, \cite{SP11, SPV18, SPD17, SPM13}. We show this approach leads to infinite time blow-up of the predator population. We propose an alternate model in which the additional food induces predator competition. Analysis herein indicates that there are threshold values $c^{*}_{1} < c^{*}_{2} < c^{*}_{3}$ of the competition parameter $c$, s.t. when $c < c^{*}_{1}$, the pest free state is globally stable, when $c^{*}_{2} < c < c^{*}_{3}$, bi-stability is possible, and when $c^{*}_{3} < c$, up to three interior equilibriums could exist. As $c$ and $\xi$-$c$ are varied, standard co-dimension one and co-dimension two bifurcations are observed. The recent dynamical systems literature involving predator competition, report several non-standard bifurcations such as the saddle-node-transcritical bifurcation (SNTC) occurring in co-dimension two \cite{KSV10, BS07}, and cusp-transcritical bifurcation (CPTC) in co-dimension three, \cite{D20, BS07}. We show that in our model structural symmetries can be exploited to construct a SNTC in co-dimension two, and a CPTC also in co-dimension two. We further use these symmetries to construct a novel pitchfork-transcritical bifurcation (PTC) in co-dimension two, thus explicitly characterizing a new organizing center of the model. Dynamics such as homoclinic orbits, concurrently occurring limit cycles, and competition driven Turing patterns are also observed. Our findings indicate that increasing additional food in predator-pest models, can hinder bio-control, contrary to some of the literature. However, additional food that also induces predator competition, leads to novel bio-control scenarios, and complements the work in \cite{H21, B98, K04, D20, BS07, VH19}., Comment: 30 pages

The soybean aphid, Aphis glycines (Hemiptera: Aphididae), is an invasive pest that can cause severe yield loss to soybeans in the northcentral United States. A tactic to counter this pest is the use of aphid-resistant soybean varieties. However, the occurrence of virulent biotypes can alter plant physiology and impair the use of this management strategy. Soybean aphids can alter soybean physiology primarily by two mechanisms, feeding facilitation and the obviation of resistance, favoring subsequent colonization by additional conspecifics. We developed a non-local, differential equation population model, to explore the dynamics of these biological mechanisms on soybean plants co-infested with virulent and avirulent aphids. We then use demographic parameters from laboratory experiments to perform numerical simulations via the model. These simulations successfully mimic various aphid dynamics observed in the field. Our model showed an increase in colonization of virulent aphids increases the likelihood that aphid-resistance is suppressed, subsequently increasing the survival of avirulent aphids, producing an indirect, positive interaction between the biotypes. These results suggest the potential for a "within plant" refuge that could contribute to the sustainable use of aphid resistant soybeans., Comment: 15 pages, 3 figures, 2 tables, Accepted in Journal of Economic Entomology

A deterministic two-species predator-prey model with prey herd behavior is considered incorporating mutual interference and the effect of fear. We provide guidelines to the dynamical analysis of biologically feasible equilibrium points. We give conditions for the existence of some local and global bifurcations at the coexistence equilibrium. We also show that fear can induce extinction of the prey population from a coexistence zone in finite time. Our numerical simulations reveal that varying the strength of fear of predators with suitable choice of parameters can stabilize and destabilize the coexistence equilibrium solutions of the model. Additionally, we discuss the outcome of introducing a constant harvesting effort to the predator population in terms of changing the dynamics of the system, in particular, from finite time extinction to stable coexistence. Furthermore, we perform extensive numerical experiments to visualize the dynamical behavior of the model and substantiate the results we obtained., Comment: 20 pages, 24 figures

We investigate the classical two species ODE and PDE Lotka-Volterra competition models, where one of the competitors could potentially go extinct in finite time. We show that in this setting, classical theories and intuitions do not hold, and various counter intuitive dynamics are possible. In particular, the weaker competitor could avoid competitive exclusion, and the slower diffuser may not win. Numerical simulations are performed to verify our analytical findings., Comment: 21 pages, 27 figures

The Trojan Y Chromosome strategy (TYC) is a genetic biocontrol strategy designed to alter the sex ratio of a target invasive population by reducing the number of females over time. Recently an alternative strategy is introduced, that mimics the TYC strategy by harvesting females whilst stocking males . We consider the FHMS strategy, with a weak Allee effect, and show that the extinction boundary need note be hyperbolic. To the best of our knowledge, this is the first example of a non-hyperbolic extinction boundary in mating models, structured by sex. Next, we consider the spatially explicit model and show that the weak Allee effect is both sufficient and necessary for Turing patterns to occur. We discuss the applicability of our results to large scale biocontrol, as well as compare and contrast our results to the case with a strong Allee effect.

Mutual interference and prey refuge are important drivers of predator-prey dynamics. The "exponent" or degree of mutual interference has been under much debate in theoretical ecology. In the present work, we investigate the interplay of the mutual interference exponent, on the behavior of a predator-prey model with a generalized Holling type functional response. We investigate stability properties of the system and derive conditions for the occurrence of saddle-node and Hopf-bifurcations. A sufficient condition for extinction of the prey species has also been derived for the model. In addition, we investigate the effect of a prey refuge on the population dynamics of the model and derive conditions for the prey refuge that would yield persistence of populations. We provide additional verification our analytical results via numerical simulations. Our findings are in accordance with classical experimental results in ecology by Gauss G. F (1934), that show that extinction of predator and prey populations is possible in a finite time period - but that bringing in refuge can effectively cause persistence., Comment: 28 pages, 27 figures

Classical language dynamics explains language shift as a process in which speakers adopt a higher status language in lieu of a lower status language. This is well documented with English having out-competed languages such as Scottish Gaelic, Welsh and Mandarin. The 1961-1991 Indian censuses report a sharp increase in Hindi/English Bilinguals, suggesting that English is on the rise in India - and is out-competing Hindi. However, the 1991 - 2011 data shows that Bilingual numbers have saturated, while Monolingual Hindi speakers continue to rise exponentially. To capture this counter-intuitive dynamic, we propose a novel language dynamics model of interaction between Monolingual Hindi speakers and Hindi/English Bilinguals, which captures the Indian census data of the last 50 years with near perfect accuracy, outperforming the best known language dynamics models from the literature. We thus provide a first example of a lower status language having out competed a higher status language., Comment: 24 pages, 6 figures, 6 tables

The Trojan Y Chromosome Strategy (TYC) is an extremely well investigated biological control method for controlling invasive populations with an XX-XY sex determinism. In \cite{GP12, WP14} various dynamical properties of the system are analyzed, including well posedness, boundedness of solutions, and conditions for extinction or recovery. These results are derived under the assumption of positive solutions. In the current manuscript, we show that if the introduction rate of trojan fish is zero, under certain large data assumptions, negative solutions are possible for the male population, which in turn can lead to finite time blow-up in the female and male populations. A comparable result is established for \emph{any} positive initial condition if the introduction rate of trojan fish is large enough. Similar finite time blow-up results are obtained in a spatial temporal TYC model that includes diffusion. Lastly, we investigate improvements to the TYC modeling construct that may dampen the mechanisms to the blow-up phenomenon or remove the negativity of solutions. The results draw into suspect the reliability of current TYC models under certain situations., Comment: 18 pages, 8 figures

The Trojan Y Chromosome (TYC) Strategy is a promising eradication method that attempts to manipulate the female to male ratio to promote the reduction of the population of an invasive species. The manipulation stems from an introduction of sex-reversed males, called supermales, into an ecosystem. The offspring of the supermales is guaranteed to be male. Mathematical models have shown that the population can be driven to extinction with a continuous supply of supermales. In this paper, a new model of the TYC strategy is introduced and analyzed that includes two important modeling characteristics, that are neglected in all previous models. First, the new model includes intraspecies competition for mates. Second, a strong Allee effect is included. Several conclusions about the strategy via optimal control are established. These results have large scale implications for the biological control of invasive species., Comment: 18 pages, 7 figures

The Trojan Y Chromosome Strategy (TYC) is a promising eradication method for biological control of non-native species. The strategy works by manipulating the sex ratio of a population through the introduction of \textit{supermales} that guarantee male offspring. In the current manuscript, we compare the TYC method with a pure harvesting strategy. We also analyze a hybrid harvesting model that mirrors the TYC strategy. The dynamic analysis leads to results on stability, global boundedness of solutions and bifurcations of the model. Several conclusions about the different strategies are established via optimal control methods. In particular, the results affirm that either a pure harvesting or hybrid strategy may work better than the TYC method at controlling an invasive species population., Comment: 37 pages, 11 figures

Blue mussels (Mytilus edulis) are an important keystone species that have been declining in the Gulf of Maine. This could be attributed to a variety of complex factors such as indirect effects due to invasion by epibionts, which remains unexplored mathematically. Based on classical optimal foraging theory and anti-fouling defense mechanisms of mussels, we derive an ODE model for crab-mussel interactions in the presence of an invasive epibiont, Didemnum vexillum. The dynamical analysis leads to results on stability, global boundedness and bifurcations of the model. Next, via optimal control methods we predict various ecological outcomes. Our results have key implications for preserving mussel populations in the advent of invasion by non-native epibionts. In particular they help us understand the changing dynamics of local predator-prey communities, due to indirect effects that epibionts confer., Comment: 33 pages, 24 figures

Biological control, the use of predators and pathogens to control target pests, is a promising alternative to chemical control. It is hypothesized that the introduced predators efficacy can be boosted by providing them with an additional food source. The current literature \cite{SP11} claims that if the additional food is of sufficient constant quantity and quality then pest eradication is possible in \emph{finite} time. The purpose of the current manuscript is to show that to the contrary, pest eradication is \emph{not possible} in finite time, for \emph{any} quantity and quality of constant additional food. However for a density dependent quantity of additional food, we show pest eradication in finite time is indeed possible. Our results have large scale implications for the effective design of biological control methods involving additional food.

In the current manuscript, an attempt has been made to understand the dynamics of a time-delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis type functional responses for large initial data. In \cite{RK15}, we have seen that the model does possess globally bounded solutions, for small initial conditions, under certain parametric restrictions. Here, we show that actually solutions to this model system can blow-up in finite time, for large initial condition, \emph{even} under the parametric restrictions derived in \cite{RK15}. We prove blow-up in the delayed model, as well as the non delayed model, providing sufficient conditions on the largeness of data, required for finite time blow-up. Numerical simulations show, that actually the initial data does not have to be very large, to induce blow-up. The spatially explicit system is seen to possess Turing instability. We have also studied Hopf-bifurcation direction in the spatial system, as well as stability of the spatial Hopf-bifurcation using the central manifold theorem and normal form theory.

Systems of reaction-diffusion equations are commonly used in biological models of food chains. The populations and their complicated interactions present numerous challenges in theory and in numerical approximation. In particular, self-diffusion is a nonlinear term that models overcrowding of a particular species. The nonlinearity complicates attempts to construct efficient and accurate numerical approximations of the underlying systems of equations. In this paper, a new nonlinear splitting algorithm is designed for a partial differential equation that incorporates self-diffusion. We present a general model that incorporates self-diffusion and develop a numerical approximation. The numerical analysis of the approximation provides criteria for stability and convergence. Numerical examples are used to illustrate the theoretical results.

An interesting conundrum in biological control questions the efficiency of generalist predators as biological control agents. Theory suggests, generalist predators are poor agents for biological control, primarily due to mutual interference. However field evidence shows they are actually quite effective in regulating pest densities. In this work we provide a plausible answer to this paradox. We analyze a three species model, where a generalist top predator is introduced into an ecosystem as a biological control, to check the population of a middle predator, that in turn is depredating on a prey species. We show that the inclusion of predator interference alone, can cause the solution of the top predator equation to blow-up in finite time, while there is global existence in the no interference case. This result shows that interference could actually cause a population explosion of the top predator, enabling it to control the target species, thus corroborating recent field evidence. Our results might also partially explain the population explosion of certain species, introduced originally for biological control purposes, such as the cane toad (\emph{Bufo marinus}) in Australia, which now functions as a generalist top predator. We also show both Turing instability and spatio-temporal chaos in the model. Lastly we investigate time delay effects.

The Allee effect is an important phenomenon in population biology characterized by positive density dependence, that is a positive correlation between population density and individual fitness. However, the effect is not well studied in multi-level trophic food chains. We consider a ratio dependent spatially explicit three species food chain model, where the top predator is subjected to a strong Allee effect. We show the existence of a global attractor for the the model, that is upper semicontinuous in the Allee threshold parameter $m$. To the best of our knowledge this is the first robustness result, for a spatially explicit three species food chain model with an Allee effect. Next, we numerically investigate the decay rate to a target attractor, that is when $m=0$, in terms of $m$. We find decay estimates that are $\mathcal{O}(m^{\gamma})$, where $\gamma$ is found explicitly. Furthermore, we prove various overexploitation theorems for the food chain model, showing that overexploitation has to be driven by the middle predator. In particular overexploitation is not possible without an Allee effect in place. We also uncover a rich class of Turing patterns in the model which depend significantly on the Allee threshold parameter $m$. Our results have potential applications to trophic cascade control, conservation efforts in food chains, as well as Allee mediated biological control.

Cannibalism, which is the act of killing and at least partial consumption of conspecifics, is ubiquitous in nature. Mathematical models have considered cannibalism in the predator primarily, and show that predator cannibalism in two species ODE models provides a strong stabilizing effect. There is strong ecological evidence that cannibalism exists among prey as well, yet this phenomenon has been much less investigated. In the current manuscript, we investigate both the ODE and spatially explicit forms of a Holling-Tanner model, with ratio dependent functional response. We show that cannibalism in the predator provides a stabilizing influence as expected. However, when cannibalism in the prey is considered, we show that it cannot stabilise the unstable interior equilibrium in the ODE case, but can destabilise the stable interior equilibrium. In the spatially explicit case, we show that in certain parameter regime, prey cannibalism can lead to pattern forming Turing dynamics, which is an impossibility without it. Lastly we consider a stochastic prey cannibalism rate, and find that it can alter both spatial patterns, as well as limit cycle dynamics.

In this work we consider a four compartment Brusselator system. The reaction terms of this system are of non constant sign, thus components of the solution are not bounded apriori, and functional means to derive apriori bounds will fail. We prove global existence of solutions, via construction of an appropriate lyapunov functional. Furthermore due to the sign changing nonlinearities, the asymptotic sign condition is also not satisfied, causing further difficulties in proving the existence of global attractor. These difficulties are circumvented via the use of the lyapunov functional constructed along with the use of the uniform gronwall lemma. We are able to prove the existence of an attractor for the system, improving previous results in the literature.The Hausdorff and fractal dimensions of the attractor are also shown to be finite.In particular we derive a lower bound on the Hausdorff dimension of the global attractor. We use numerical simulations,as well as numerical attractor reconstruction methods via nonlinear time series analysis, to validate our results.

In this work we develop and analyze a mathematical model of biological control to prevent or attenuate the explosive increase of an invasive species population in a three-species food chain. We allow for finite time blow-up in the model as a mathematical construct to mimic the explosive increase in population, enabling the species to reach "disastrous" levels, in a finite time. We next propose various controls to drive down the invasive population growth and, in certain cases, eliminate blow-up. The controls avoid chemical treatments and/or natural enemy introduction, thus eliminating various non-target effects associated with such classical methods. We refer to these new controls as "ecological damping", as their inclusion dampens the invasive species population growth. Further, we improve prior results on the regularity and Turing instability of the three-species model that were derived in earlier work. Lastly, we confirm the existence of spatio-temporal chaos.

While language competition models of diachronic language shift are increasingly sophisticated, drawing on sociolinguistic components like variable language prestige, distance from language centers and intermediate bilingual transitionary populations, in one significant way they fall short. They fail to consider contact-based outcomes resulting in mixed language practices, e.g. outcome scenarios such as creoles or unmarked code switching as an emergent communicative norm. On these lines something very interesting is uncovered in India, where traditionally there have been monolingual Hindi speakers and Hindi/English bilinguals, but virtually no monolingual English speakers. While the Indian census data reports a sharp increase in the proportion of Hindi/English bilinguals, we argue that the number of Hindi/English bilinguals in India is inaccurate, given a new class of urban individuals speaking a mixed lect of Hindi and English, popularly known as "Hinglish". Based on predator-prey, sociolinguistic theories, salient local ecological factors and the rural-urban divide in India, we propose a new mathematical model of interacting monolingual Hindi speakers, Hindi/English bilinguals and Hinglish speakers. The model yields globally asymptotic stable states of coexistence, as well as bilingual extinction. To validate our model, sociolinguistic data from different Indian classes are contrasted with census reports: We see that purported urban Hindi/English bilinguals are unable to maintain fluent Hindi speech and instead produce Hinglish, whereas rural speakers evidence monolingual Hindi. Thus we present evidence for the first time where an unrecognized mixed lect involving English but not "English", has possibly taken over a sizeable faction of a large global population., Comment: This paper has been withdrawan as the model has now been modified and the existing model has some errors

In this work we use functional methods to prove the boundedness and global existence of solutions for a class of strongly coupled parabolic systems. We apply the results to deduce the global existence of solutions for a classic Shigesada-Kawasaki-Teramoto (SKT) type model, for an extended range of the self-diffusion and cross-diffusion coefficients than those available in the current literature. We provide numerical simulations in 2D, via a spectral Galerkin method to verify our global existence results, as well as to visualize the dynamics of the system.

In [Aziz-Alaoui, 2002] a three species ODE model, based on a modified Leslie-Gower scheme is investigated. It is shown that under certain restrictions on the parameter space, the model has bounded solutions for all positive initial conditions, which eventually enter an invariant attracting set. We show that this is not true. To the contrary, solutions to the model can blow up in finite time, even under the restrictions derived in [Aziz-Alaoui, 2002], if the initial data is large enough. We also prove similar results for the spatially extended system. We validate all of our results via numerical simulations., Comment: 10 pages, 4 figures

We consider nonlinear parabolic SPDEs of the form $\partial_t u=-(-\Delta)^{\alpha/2} u + b(u) +\sigma(u)\dot w$, where$\dot w$ denotes space-time white noise. The functions $b$ and $\sigma$ are both locally Lipschitz continuous. Under some suitable conditions on the parameters of these SPDEs, we show that the second moment of their solutions blow up in finite time. This complements recent works of Khoshnevisan and his coauthors; see for instance Foondun and Khoshnevisan (2009), Foondun and Khoshnevisan (to appear, 2012) and Conus, Joseph, and Khoshnevisan, as well as those of Chow (Chow, 2009, and Chow, 2011). Furthermore, upon comparing our stochastic equations with their deterministic counterparts, we find that our results indicates that the presence of noise might affect the occurrence of blow-up.