THE EFFECT OF FEAR ON TWO SPECIES COMPETITION Vaibhava Srivastava1 Eric M. Takyi2and Rana D. Parshad1 1Department of Mathematics

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THE EFFECT OF “FEAR” ON TWO SPECIES COMPETITION

Vaibhava Srivastava 1, Eric M. Takyi 2and Rana D. Parshad 1

1)Department of Mathematics,

Iowa State University,

Ames, IA 50011, USA.

2)Department of Mathematics and Computer Science,

Ursinus College,

Collegeville, PA 19426, USA.

Abstract. 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 moti-

vations for the same. In this work we consider the classic two species ODE

and PDE Lokta-Volterra competition models, where 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 com-

petition, 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 L1re-

strictions 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.

1. Introduction

Fear, is deﬁned as,

An unpleasant emotion caused by the belief that someone or some-

thing is dangerous [77].

It is a complex emotion, that is critical as a safety measure, and can trigger

the “ﬁght or ﬂight” response [2] - in particular it can change the way one acts,

even when there is no threat present [24]. In predator-prey systems, this is most

naturally observed among prey, due to their perceived threat of depredation [10].

This perception can lead to non-consumptive eﬀects or trait-mediated interactions,

which are behavioral, morphological or physiological changes in prey phenotype, due

to this threat [23,10]. Such eﬀects are known to strongly inﬂuence predator-prey

dynamics [21]. From a mathematical viewpoint, the eﬀects of fear in predator-

prey systems has been intensely investigated since the seminal work of Brown et.

al. [28], where optimal foraging theory is extended to consider a game theoretic

setup, played out by predator and prey, exhibiting stealth and fear, in which an

animal follows a map or a “landscape of fear” (LOF), which describes its predation

risk while it navigates the physical landscape. In recent work, Wang et. al. [50],

model fear of depredation, as a (predator) density dependent eﬀect, that negatively

eﬀects the prey population. In essence, the prey’s growth rate is modeled as a

arXiv:2210.10280v1 [q-bio.PE] 19 Oct 2022

2 FEAR EFFECT

monotonically decreasing function of predator density. Dynamically, a key ﬁnding

in [50] is that under the parametric restrictions of a Hopf bifurcation, an increase

in the fear parameter (and prey’s birth rate parameter) can alter the direction

of a Hopf bifurcation from supercritical to subcritical. Thus, fear enables both

supercritical and subcritical Hopf bifurcations, contrary to only the supercritical

bifurcations found in classical predator-prey systems. In essence, the fear eﬀect

can change the fundamental cylical patterns of predator-prey dynamics, leading to

large scale ecological consequences [12].

These results have since initiated a host of activities in diverse ecological sce-

narios such as when refuges are present [59,51], when the prey has tendencies to

avoid predators [52], or when the predators responses are inﬂuenced by interfer-

ence pressures, for instance, via a Beddington-DeAngelis functional response [60].

Various works have considered the fear eﬀect in case of group defense by the prey

[65,64]. It has been investigated in the context of cooperative and competitive sys-

tems within the larger predator-prey context. These include the fear eﬀect when

predators are cooperating [55] in the hunting process, or when they are hunting for

competing prey [14]. These eﬀects have been investigated in the three and multi-

species settings as well [53,68] where fear can damp population explosions [71].

Various authors have considered the fear eﬀect in a stochastic setting [70] as well

as a spatially explicit setting, in the context of taxis type movements, as well as

pattern formation [56,62]. It can also lead to chaotic dynamics [67]. However,

the eﬀect of fear has been far less investigated in classical monotone systems, such

as purely cooperative or competitive two species systems - that are outside the

predator-prey setting.

Competition among two species, typically modeled via the Lotka–Volterra com-

petition model and its variants have been intensely investigated in the last few

decades. These models take into account growth and inter/intraspeciﬁc competi-

tion [41], and predict well-observed states in biology of co-existence, competitive

exclusion of one competitor, and bi-stability, and ﬁnd diverse applications in ecol-

ogy and invasion science [13,15,3,6,1]. There are several ecological motivations

for competitors being fearful of each other. This is perhaps most naturally seen

to occur with intraguild predation - a widespread phenomenon in many food webs,

where competitors will kill and consume each other [7]. Recent evidence of non-

consumptive eﬀects exerted by intraguild predator mites (Blattisocius dentriticus)

on their competitor (Neoseiulus cucumeris) show this can be an important factor

in determining food web dynamics in biological control [73,4]. However, there is

strong evidence for fear in purely competitive two species systems without preda-

tory eﬀects. Barred owls (Strix varia) are a species of owl, native to eastern North

America. They have expanded their range westward over the last century and are

considered invasive in western North America. Currently, their range overlaps with

the spotted owl (Strix occidentalis), which is native to the north west and western

North America. This has resulted in intense competition between the two species

[72]. Barred owls exert a strong negative inﬂuence on spotted owls, threatening

their possible competitive exclusion [75]. Field observations report frequent barred

owl attacks on spotted owls, and even on surveyors imitating spotted owl calls

[74]. There is also evidence of barred owls aggressively chasing spotted owls out of

FEAR EFFECT 3

shared habitat - but not the opposite [76]. Such evidence clearly motivates consid-

ering fear type dynamics into a purely competitive two species model where one of

the competitors is fearful of the other.

There are also several socio-economic-political settings, where pure competitors

may be fearful of each other. Small/new businesses may be fearful of large busi-

nesses, due to their already large market share [43]. But large business may also

be fearful of small local businesses, due to their familiarity with local nuances, that

may yield competitive advantage at a small local scale [42]. Fear is also conceiv-

able among two competing political parties, where the weaker party on a national

scale, may have a stronger voter bank at a regional scale [18]. Or perhaps two war-

ring drug cartels, where the weaker cartel has certain local/territorial strongholds

[19,44] - within which they might be able to induce fear among the stronger cartel

[19]. Such phenomenon becomes even more interesting in the spatially explicit case

where this fear could be heterogeneous in the spatial domain of interest. This con-

nects back to the LOF concept, where the fear function is essentially the map that

describes how the fear levels change as a species disperses over a physical landscape.

Motivated by all of the aﬀore mentioned sociopolitical, economic as well as eco-

logical settings, the current manuscript considers the eﬀect of fear in a competitive

two species system. We restrict our analysis to the case where only one of the

competitors is fearful of the other. Our investigations show that:

•Suﬃciently large fear can change a situation of competitive exclusion, to a

strong competition type scenario, where there is bi-stability between bound-

ary equilibrium. See Fig. 5(C). Dynamically, this occurs via a transcritical

bifurcation. This is shown via Lemma 2.19, see Fig. 9.

•Fear in a certain parametric regime can change a situation of competitive

exclusion to bi-stability between boundary equilibrium and interior equilib-

rium, see Fig. 4and Fig. 5(B). Dynamically, this occurs via a saddle-node

bifurcation. This is shown via Lemma 2.18, see Fig. 8. This is in sharp

contrast with classical competition theory, where bi-stability occurs only

between boundary equilibriums.

•Suﬃciently large fear can change a situation of weak competition to a com-

petitive exclusion type scenario. This is shown via Lemma 2.5, see Fig. 2.

•Fear cannot qualitatively change a strong competition type scenario. This

is shown via Lemma 2.8, see Fig. 3. Also, fear cannot produce periodic

orbits. This is demonstrated via Lemma 2.10.

•In the spatially explicit setting, comparison theory is used to determine

point-wise restrictions on the fear functions such that competitive exclu-

sion or strong competition type dynamics abounds. These are shown via

Theorem 3.8, Theorem 3.10 and Theorem 3.15, see Figs. [10,11,13].

•In the spatially explicit setting, fear can change a situation of weak com-

petition to a competitive exclusion type scenario, for fear functions with

certain L1restrictions. This is shown via Theorem 3.13 and Lemma 3.14,

see Figs. [15,16,17]. In particular the fear functions need not lie uniformly

above the critical fear levels derived in the ODE case via Lemma 2.5.

•Various heterogeneous fear functions are constructed to demonstrate these

results numerically, see Fig. 18b. Applications of these to ecological as well

as socio-political settings are discussed in section 4.

4 FEAR EFFECT

2. The ODE case

2.1. Model formulation. Consider the classical two species Lotka-Volterra ODE

competition model,

(1) 









dt =u(a1−b1u−c1v),

dt =v(a2−b2v−c2u),

where uand vare the population densities of two competing species, a1and a2are

the intrinsic (per capita) growth rates, b1and b2are the intraspeciﬁc competition

rates, c1and c2are the interspeciﬁc competition rates. All parameters considered

are positive. The dynamics of this system are well studied [20]. We recap these

brieﬂy,

•E0= (0,0) is always unstable.

•Eu= (a1

b1,0) is globally asymptotically stable if a1

>max b1

,c1

b2.

Herein uis said to competitively exclude v.

•Ev= (0,a2

b2) is globally asymptotically stable if a1

<min b1

,c1

b2. Herein

vis said to competitively exclude u.

•E∗=a1b2−a2c1

b1b2−c1c2,a2b1−a1c2

b1b2−c1c2exists when b1b2−c1c26= 0. The positivity of

the equilibrium holds if c2

b1<a2

a1<b2

c1and is globally asymptotically stable

if b1b2−c1c2>0. This is said to be the case of weak competition.

•If b1b2−c1c2<0, then E∗=a1b2−a2c1

b1b2−c1c2,a2b1−a1c2

b1b2−c1c2is unstable as a saddle.

In this setting, one has initial condition dependent attraction to either

Eu(a1

b1,0) or Ev(0,a2

b2). This is the case of strong competition.

We proceed by considering the eﬀects of fear on the classical model (1), when

one of the competitors is fearful of the other.

2.2. The case of vfearing u.We consider the case of the competitor vbeing

fearful of u. Thus in the classical model (1), we model the fear eﬀect as in [50],

where the growth rate of the fearful competitor v, is not constant but rather density

dependent. Essentially, the growth rate is decreased by a factor ≈1

1+ku , where

k≥0 is a fear coeﬃcient. Thus a higher density of the competitor uincreases the

fear in v. When k= 0, the assumption is there is no fear and one recovers the

classical model (1). If fear is present, we obtain the following ODE model for two

competing species uand v, where vis fearful of u.

dt =a1u−b1u2−c1uv,

dt =a2v

1 + ku −b2v2−c2uv.

(2)

2.2.1. Existence. The nullclines associated with the problem (2) are

u(a1−b1u−c1v) = 0 and va2

1 + ku −b2v−c2u.

FEAR EFFECT 5

Hence, the boundary equilibrium points are obtained by substituting u= 0 and

v= 0 in the above equations of the nullclines, respectively. Denote the boundary

equilibrium points as b

E1= (0,0), b

E2= (a1

b1,0) and b

E3= (0,a2

b2).

For the interior equilibrium, substitute u∗=a1

b1−c1

b1v∗in the second nullcline

equation, i.e.,

1 + ka1

b1−c1

b1v∗−b2v∗−c2a1

b1−c1

v∗= 0.

On simpliﬁcation, we have that v∗solves a quadratic equation of the form

A(v∗)2+Bv∗+C= 0, where

A=c1k(b2b1−c1c2),

B=b1(c1c2−b2b1)−a1k(b2b1−2c1c2),

C=b1(a2b1−a1c2)−a2

1c2k.

(3)

Let

(4) v∗

1,2=−B±√B2−4AC

be the two roots of above qudratic equation. WLOG assume v∗

1< v∗

2.Moreover,

consider the following parametric restriction

b1(a1c2−a2b1) + a2

1c2k < h(b2b1−c1c2)(2a1k−b1)−a1kc1c2ia1

.(5)

We can prove the existence of a positive equilibrium point b

E4with the choice of

speciﬁc parameters. Let us use Descartes’s rule of sign to establish some suﬃcient

conditions for the existence of one or two positive equilibrium points.

Two positive equilibrium points: Under the assumption A > 0, B < 0 and C > 0,

i.e., b2b1>2c1c2> c1c2and k < 1

1c2b2

1a2−a1c2b1,we have two positive roots.

In order to claim that these two roots correspond to two positive interior equilibria,

we need some extra assumption given by:

v∗

1< v∗

2:= −B+√B2−4AC

2A<a1

=⇒ −C < (Aa1

+B)a1

Hence, if b1(a1c2−a2b1) + a2

1c2k < [(b2b1−c1c2)(2a1k−b1)−a1kc1c2]a1

c1,we have

two positive interior equilibrium points b

E4= (u∗

i, v∗

i) for i= 1,2.

One positive equilibrium point: Under the assumption A > 0, B < 0, i.e.,

b2b1>2c1c2> c1c2, we have at least one positive root of the quadratic equa-

tion. If C < 0, which is k > 1

1c2b2

1a2−a1c2b1along with (5) gives existence

of one positive equilibrium point b

E4= (u∗, v∗). Moreover, if C > 0, which is

k < 1

1c2b2

1a2−a1c2b1, and v∗

1<a1

c1and v∗

2>a1

c1, then we have existence of one

positive equilibrium point b

E4= (u∗, v∗).

We formulate all these restrictions as an existence theorem:

Theorem 2.1. For the given ODE system (2), we always have three boundary

equilibrium points, namely b

E1:= (0,0),b

E2:= (a1

b1,0) and b

E3:= (0,a2

b2). For the

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THEEFFECTOF\FEAR"ONTWOSPECIESCOMPETITIONVaibhavaSrivastava1,EricM.Takyi2andRanaD.Parshad11)DepartmentofMathematics,IowaStateUniversity,Ames,IA50011,USA.2)DepartmentofMathematicsandComputerScience,UrsinusCollege,Collegeville,PA19426,USA.Abstract.Non-consumptiveeectssuchasfearofdepredation,canstrongl...

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THE EFFECT OF FEAR ON TWO SPECIES COMPETITION Vaibhava Srivastava1 Eric M. Takyi2and Rana D. Parshad1 1Department of Mathematics

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