arXiv subjects

Rana D. Parshad

Publications and source records attributed to Rana D. Parshad.

At least 37 records · Page 2Linked to original sources

arXiv2022

The effect of "fear" on two species competition

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.

q-bio.PE↗

arXiv2022

Additional food causes predator "explosion" -- unless the predators compete

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 $ξ> ξ_{critical}$, \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 $ξ$-$c$ 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}.

q-bio.PE↗

arXiv2021

Fear-driven extinction and (de)stabilization in a predator-prey model incorporating prey herd behavior and mutual interference

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.

math.DS↗

arXiv2020

Some "counterintuitive" results in two species competition

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.

math.AP↗

arXiv2020

Mimicking the TYC strategy: Weak Allee effects, and a nonhyperbolic extinction boundary

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.

q-bio.PE↗

arXiv2020

Dynamics of a predator-prey model with generalized Holling type functional response and mutual interference

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.

math.DS↗

arXiv2020

The "higher" status language does not always win: The fall of English in India and the rise of Hindi

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.

physics.soc-ph↗

arXiv2019

A Comparison of the Trojan Y Chromosome Strategy to Harvesting Models for Eradication of Non-Native Species

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.

math.OC↗

arXiv2019

Large and Small Data Blow-Up Solutions in the Trojan Y Chromosome Model

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.

math.DS↗

arXiv2019

Optimal Control and Analysis of a Modified Trojan Y-Chromosome Strategy

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.

q-bio.PE↗

arXiv2018

The effects of invasive epibionts on crab-mussel communities: a theoretical approach to understand mussel population decline

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.

q-bio.PE↗

arXiv2018

A Constant Introduction of Additional Food Does Not Yield Pest Eradication in Finite Time

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.

q-bio.PE↗

arXiv2015

What is India speaking: The "Hinglish" invasion

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.

cs.CL↗

arXiv2015

Finite time Blow up in a population model with competitive interference and time delay

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.

math.AP↗

arXiv2015

Predator interference effects on biological control: The "paradox" of the generalist predator revisited

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.

q-bio.PE↗

arXiv2015

Long time dynamics of a three-species food chain model with Allee effect in the top predator

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 $\mathcal{O}(m^γ)$, where $γ$ 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.

math.AP↗

arXiv2015

Prey cannibalism alters the dynamics of Holling-Tanner type predator-prey models

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.

q-bio.PE↗

arXiv2015

Global Existence and Long time dynamics of a four compartment Brusselator Type system

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.

math.DS↗