AGeneralModelofPopulationDynamicsAccountingfor ...downloads.hindawi.com/journals/complexity/2020/7961327.pdfResearchArticle AGeneralModelofPopulationDynamicsAccountingfor MultipleKindsofInteraction

Research ArticleA General Model of Population Dynamics Accounting forMultiple Kinds of Interaction

Luciano Stucchi12 Juan Manuel Pastor2 Javier Garcıa-Algarra 3 and Javier Galeano 2

1Universidad Del Pacıfico Lima Peru2Complex Systems Group E T S I A A B Universidad Politecnica de Madrid Madrid Spain3Department of Engineering Centro Universitario U-TAD Las Rozas Spain

Correspondence should be addressed to Javier Galeano javiergaleanoupmes

Received 13 May 2020 Accepted 22 June 2020 Published 24 July 2020

Guest Editor Tongqian Zhang

Copyright copy 2020 Luciano Stucchi et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Population dynamics has been modelled using differential equations almost since Malthus times more than two centuries agoBasic ingredients of population dynamics models are typically a growth rate a saturation term in the form of Verhulstrsquos logisticbrake and a functional response accounting for interspecific interactions However intraspecific interactions are not usuallyincluded in the equations e simplest models use linear terms to represent a simple picture of the nature meanwhile torepresent more complex landscapes it is necessary to include more terms with a higher order or that are analytically morecomplex e problem to use a simpler or more complex model depends on many factors mathematical ecological orcomputational To address it here we discuss a new model based on a previous logistic-mutualistic model We have generalizedthe interspecific terms (for antagonistic and competitive relationships) and we have also included new polynomial terms toexplain any intraspecific interaction We show that by adding simple intraspecific terms new free-equilibrium solutions appeardriving a much richer dynamics ese new solutions could represent more realistic ecological landscapes by including a newhigher order term

1 Introduction

In the times of the coronavirus many news on television andmagazines try to explain how the size of the infected pop-ulation evolves showing exponential plots of the infectedpopulations over time ese communications try to predictthe time evolution of the size of this population in the futureBehind these predictions there is always a differentialequationmodelese polynomial models have linear termsbut to account for more complex interactions they can addhigher order terms as quadratic cubic or even analyticallymore complex functions such as decreasing hyperbolictermse problem of choosing a complex or a simple modeldepends on the balance between properly representingnature and being able to understand the model response Inmany cases the simplest model may be enough to under-stand the benchmarks in the big picture but sometimes we

need more complexity to represent significant aspects of ourproblem and therefore we need more complex and moredifficult models Finding the balance between simple andcomplex is a tricky problem but how simple or complexshould the model be Let us try to answer this question in apopulation dynamics problem

In the study of population dynamics Lotka [1] andVolterra [2] were the first ones to model trophic interactionsin order to study predator-prey relationships within two (ormore) populations

X1middot

r1 minus b12X2( 1113857X1

X2middot

minus r2 + b21X1( 1113857X2

(1)

where bij terms represent the rate of the interactions be-tween populations Xi and Xj and the ri represents theireffective growth rates In these equations signs are

HindawiComplexityVolume 2020 Article ID 7961327 14 pageshttpsdoiorg10115520207961327

incorporated to give a clear meaning to each term con-sidering all parameters as positive real numbers is simplemodel uses a linear term to represent the interaction with theenvironment and a pairwise second-order term to show theantagonistic interaction between the populations of twodifferent species It was necessary to introduce a higher orderterm to represent this interaction

Although most population dynamics models first dealtwith antagonistic relations mutualistic interactions arewidely spread eg [3 4] Garcia-Algarra et al [5] proposed alogistic-mutualistic model eir formulation was based onwriting an effective growth rate as the sum of the intrinsicgrowth rate (ri) plus the mutualistic benefit (bijXj) andassociated with them to include a saturation term for thewhole effective growth term e model was depicted as

X1middot

r1 + b12X2( 1113857X1 minus a1 + c1b12X2( 1113857X21

X2middot

r2 + b21X1( 1113857X2 minus a2 + c2b21X1( 1113857X22

(2)

e term aiX2i represents the intraspecific competition for

resources and the term cibijXjX2i plays the role of saturation

for the mutualistic benefit is model needs to reach a third-order term to prevent the unbounded growth and depicts awell behaved system with enough richness to model largeensembles of mutualistic networks and their behaviour

Other authors have addressed different strategies tointroduce the mutualistic interaction For example Dean [6]introduced an exponential dependency on the carryingcapacity K which consequently yields nonlinear terms intothe equations To avoid the unbounded growth the authorsin [7 8] proposed restrictions using a type II Hollingfunctional response is functional leaves the path of in-troducing a polynomial term with a hyperbolic function

Nowadays several studies have focused on adding higherorder terms to explain more complex ecological interactionsLetten and Stouffer [9] studied the influence of interspecificinteractions as nonadditive density-dependent terms onlyfor competitive communities Bairey et al [10] studied thenew solutions that a third species adds in pairwise inter-actions adding third degree terms with the three differentpopulations bijkXiXjXk It is well known that an increase inthe order of a polynomial term introduces new solutions tothe equations but as AlAdwani and Saavedra [11] showedthese new terms do not always produce viable solutionsfurthermore they must be free-equilibrium points and ofcourse the solutions must have an ecological meaning

Here we propose a new general model in which anyecological interaction can be included in a simple way In afirst step we generalise the model proposed by Garcia-Algarra et al [5] overcoming the restrictions of the sign ofthe parameters in a second step we reorganize the intra-specific interactions allowing for both positive and negativeinteractions and finally we introduce a third-order term tobrake any unbounded pairwise interactions

2 Methods and Materials

We define a new general model Equation (3) represents thepopulation dynamics of the species Xi driven by an effective

growth rate (first parenthesis in equation (3)) and limited bya logistic brake (second parenthesis in equation (3)) eview of the model is simple and similar to the originalVerhulst idea [12] where the low-order terms represent theincrease in the population and the high-order terms thebrake e differences with other models are in the termsincluded in the effective growth rate and logistic brake eeffective growth rate includes the vegetative growth rate riand all density-dependent pairwise interactions interspe-cific interactions bijXj(foralljne i) and intraspecific ones biiXithe logistic brake includes the logistic term due to intra-specific competition ai the interspecific intraspecific brakebijXjXi and the intraspecific ones biiX

2i

21 A New General Model including Intraspecific InteractionTerms Regarding the mutualistic model (equation (2)) weintroduce two differences first the parameters of theequation ri and bij can be positive or negative representingthe different ecological interactions and second we includethe effect of the population in its own effective growth ratejust adding the index j i in the sum of the interactionsterms so the model can be represented as

Xi

middot

Xi ri + 1113944n

j1bijXj

⎛⎝ ⎞⎠ minus ai + ci 1113944

n

j1bijXj

⎛⎝ ⎞⎠Xi⎡⎢⎢⎣ ⎤⎥⎥⎦ (3)

where the subscript i runs from 1 to n including the in-traspecific interaction (j i) With this term we are takinginto account the interaction between individuals of the samepopulations e new terms yield new solutions and a dif-ferent phase space In particular the inclusion of the termminus cibiiX

3i is key for the emergence of new solutions although

there was already a term with the same order in the mu-tualistic logistic model (equation (2)) cibijXjX

2i It can be

observed in Figure 13 in Supplementary Materials (availablehere) We explain all details about the number of solutions inAppendix in Supplementary Materials (available here)

Generally in the literature of populations dynamics theintraspecific interactions have been introduced only as alogistic brake minus aiXi representing a growth limit due toresource sharing In our model the term bii can representany kind of intraspecific interaction from beneficial namelycooperation to harmful interactions such as competition oreven cannibalism Even though the logistic term minus aiXi canbe seen as the result of intraspecific interactions that limit thegrowth by resource sharing and it can be included in theinteraction term biiXi we maintain the separated formu-lation for the sake of comparison with the equation withoutthis new term

In fact there are abundant examples of different in-traspecific behaviours in the literature such as thosementioned above Cooperation is well known among socialand eusocial species [13] and benefits of cooperative be-haviour have been consistently reported especially foreusocial animals [14] On the other way in nature we canfind different types of competition among members of thesame population For example Stucchi and Figueroa [15]reported the aggressive intraspecific behaviour of the

2 Complexity

Peruvian booby which attacks their peers not by means oftaking their food but for the sake of being around their nestIn the same way adult boobies show little tolerance forpigeons that are not from them pecking them to death isbehaviour is well known for other territorial animals and itconceptually differs from the conventional intraspecificcompetition for resources

22 Solutions with One Population In general the equationsystem (3) cannot be solved analytically However the studyof only one population can be solved and illustrates thepossibilities of the model

Consider equation (3) for only one population eequation can be written as

_X X[(r + b middot X) minus (a + cb middot X)X] (4)

where we have removed the subscripts for simplicity Sta-tionary points where _X 0 give us the keys to understandthe behaviour of the time evolution of the population sizese trivial solution which corresponds to extinction isX⋆ 0 Now the nontrivial stationary solutions can beobtained from the following condition

r +(b minus a)X⋆

minus c middot bX⋆2

0 (5)

en the stationary solutions of equation (4) are theextinction and the solutions of equation (5)

X⋆plusmn

(b minus a) plusmn

(b minus a)2 + 4rbc

1113969

2bc (6)

In ecology we are only interested in the positive real so-lutions generally called feasible solutions To obtain these fea-sible solutions in equation (6) we need to study several cases

(i) r and b have the same sign In addition to the trivialsolution X⋆ 0 in both cases there is one positivestationary point which corresponds to the carryingcapacity of the population and the other is negativewhich is not a feasible solution

(a) In the case that both parameters are negativesrlt 0 and blt 0 the positive solution is unstableand the trivial solution is the unique stablesolution

(b) In the opposite case rgt 0 and bgt 0 the carryingcapacity is the stable solution

(ii) r and b have different signs e interesting point ofhaving a high-order term comes from the possibilityof different signs of the parameters When r and b

have different signs there are two solutions as longas the condition cle (b minus a)24|rb| is fulfilled

(a) If bgt 0 it is a necessary another condition toobtain a feasible solution that bgt a Ecologicallyspeaking this means that the term of intraspe-cific interaction overcomes the intrinsic growthdeficiency and increases the population InFigure 1(b) we plot a case with these conditionsWe obtain three fixed points initial and end

points are stable and the intermediate point isunstable is point marks the threshold pop-ulation above this value intraspecific coopera-tion moves the population to reach the carryingcapacity and below this value the populationgoes to extinction

(b) If blt 0 In this scenario the intermediate point isstable and the other solutions are unstableConsequently the intraspecific competitionsproduce a new stationary solution lower thanthe carrying capacity is behaviour has beencalled as Allee effect [16] See the example inFigure 1(b)

In Figure 1 we depict on the top _X vs X and on thebottom the temporal evolution of the population size X(t)

vs t On the left the growth rate r is negative and theintraspecific interaction coefficient b is positive e inter-mediate stationary solution plays the role of a populationthreshold because smaller communities will go extinct(population in orange in Figure 1(c)) while larger com-munities will grow to its carrying capacity (population ingreen in Figure 1(c)) On the right the growth rate r ispositive and the interaction coefficient b is negative In thiscase the carrying capacity becomes unstable and the systemevolves to the new stable intermediate solution because ofthe detrimental intraspecific interaction (both populationsin Figure 1(d)) Two examples of population evolution areplotted in each scenario where the orange and green dots inthe upper plot depict the initial condition of each evolutionin the lower plot

23 Solutions with Two Populations In the case of twopopulations the general model is written as

X1middot

X1 r1 + b11X1 + b12X2( 1113857 minus a1 + c1b11X1 + c1b12X2( 1113857X11113858 1113859

X2middot

X2 r2 + b22X2 + b21X1( 1113857 minus a2 + c2b22X2 + c2b21X1( 1113857X21113858 1113859

(7)

For two populations we also find the expected trivialsolution ie the total extinction (X⋆1 0 and X⋆2 0) andthe partial extinctions (X⋆1 0) and (0 X⋆2 ) from the fol-lowing equations

r1 + b11 minus a1( 1113857X⋆1 minus c1b11X

⋆21 0

r2 + b22 minus a2( 1113857X⋆2 minus c2b22X

⋆22 0

(8)

However as they are second-order equations there aretwo solutions of feasible partial extinctions for each pop-ulation e coexistence solutions can be obtained fromequation (7) these equations can exhibit up to 6 new sta-tionary solutions Concerning the finite stationary solutionsthe intraspecific term makes it more difficult to obtain ananalytic expression from the following equations

r1 + b11X⋆1 + b12X

⋆2( 1113857 minus a1 + c1b11X

⋆1 + c1b12X

⋆2( 1113857X⋆1 0

r2 + b22X⋆2 + b21X

⋆1( 1113857 minus a2 + c2b22X

⋆2 + c2b21X

⋆1( 1113857X⋆2 0

(9)

Complexity 3

Two out of these six solutions are new free-equilibriumpoints due to the new intraspecific terms (details in Supple-mentary Material) Even though we cannot obtain analyticexpressions for all solutions we explored different scenarios byperforming numerical simulations with different parametervalues In the next section we show how the intraspecificinteraction changes the phase space of the standard biologicalinteractions

24 Linear Stability Analysis In the next section we explorethe linear stability analysis of our system solutions

241 One Population Model To perform the linear stabilityanalysis of the stationary solutions we derive equation (4) atthe fixed points

d _X

dX

11138681113868111386811138681113868111386811138681113868X0 r (10)

d _X

dX

11138681113868111386811138681113868111386811138681113868XX⋆plusmn

r + 2(b minus a) minus 3cbX⋆

1113858 1113859X⋆

minus r minus cbX⋆2

(11)

In the trivial solution the eigenvalue is λ r and theunique stable solution is rlt 0

According to equation (11) the derivative at the (pos-itive) stationary solution X⋆ will be negative when

(b minus a) minus 2cbX⋆plusmn ∓δ lt 0 (12)

en X⋆+ is always stable and X⋆minus is unstableWhen rgt 0 and bgt 0 extinction is an unstable solution

and population rises to the carrying capacity at X⋆+ the onlypositive nontrivial solution However for rgt 0 and blt 0 iewith intraspecific competition a new stationary solutionemerges X⋆minus gtX⋆+ Now the higher solution is unstable andthe population only reaches a lower value at the stable pointX⋆+ In this case the negative intraspecific interaction resultsin a lower carrying capacity

When rlt 0 extinction is stable If blt 0 the only positivefinite solution is X⋆minus which is unstable However whenbgt agt 0 a new stable solution X⋆+ emerges at higher valuesthan X⋆minus In this scenario X⋆minus marks the threshold pop-ulation above this value intraspecific cooperationmoves thepopulation to reach the carrying capacity and below thisvalue the population goes to extinction (see Figure 1(c))

10

05

00

dX1dt

ndash05

0 10 20 30X1

40 50 60 70

(a)

02

01

00

dX1dt

ndash01

ndash02

X1

0 5 10 15 20

(b)

70

60

50

40

30

20

10

0 20 40 60 80 100t (au)

X1 [0] = 138

X1 [0] = 50

Popu

latio

n

(c)

0 20 40 60 80 100t (au)

X1 [0] = 5

X1 [0] = 15

5

10

15

20

Popu

latio

n

(d)

Figure 1 Temporal derivative (up) and population evolution (down) for one population with intraspecific interaction Negative growth rate(left) r minus 01 with positive intraspecific interaction b 0005 and c 0005 Positive growth rate (right) r 01 with negative intra-specific interaction b minus 0015 and c 005

4 Complexity

242 Two Populations Model e linear stability for thegeneralmodel (equation (3)) can be analyzed from the Jacobianmatrix at the stationary solutions Its entries are obtained from

zfi

zXi

11138681113868111386811138681113868111386811138681113868X⋆ gi X

⋆( 1113857 + bii minus ai( 1113857X

⋆i minus 2cibiiX

⋆2i minus ci 1113944

jneibijX⋆i X⋆j

zfi

zXj

111386811138681113868111386811138681113868111386811138681113868X⋆

bijX⋆i 1 minus ciX

⋆i( 1113857

(13)

where X⋆ (X⋆1 X⋆i X⋆j ) is the vector of thestationary solution

For two populations the Jacobian matrix for the totalextinction is

J 00 r1 0

0 r21113888 1113889 (14)

whose eigenvalues λ1 r1 and λ2 r2 are negative whenboth growth rates are negative For the partial extinctionsthe Jacobian matrix reads

J X⋆1 0 minus r1 minus c1b11X

⋆21 b12X

⋆1 1 minus c1X

⋆11113858 1113859

0 r2 + b21X⋆1

⎛⎝ ⎞⎠ (15)

As expected this Jacobian matrix is almost the same asthe matrix for the logistic-mutualistic model (see AppendixA in [5]) but the first entry includes the intraspecific in-teraction term minus c1b11X

⋆21 is new term makes the partial

extinction to be stable when the intraspecific interaction ispositive b11 gt 0 e same is stated for the symmetric so-lution (0 X⋆2 )

And for the nontrivial solution (X⋆1 X⋆2 ) the Jacobianmatrix is written as follows

J X⋆1 X⋆2 minus r1 minus b12X

⋆2 minus c1b11X

⋆21 b12X

⋆1 1 minus c1X

⋆11113858 1113859

b21X⋆2 1 minus c2X

⋆21113858 1113859 minus r2 minus b21X

⋆1 minus c2b22X

⋆22

⎛⎝ ⎞⎠

(16)

In this case both diagonal entries include the intra-specific term with a negative sign is means that a positiveintraspecific direct interaction enhances the stability of thisstationary solution while a negative intraspecific directinteraction contributes to destabilize it

A qualitative study of the linear stability can also bemadeby analyzing the nullclines Solving the nullclinesf1(X1 X2) 0 we obtain two solutions X1 0 as follows

g1 X1 ne 0( 1113857 r1 + b11 minus a1( 1113857X1 minus c1b11X12

minus c1X1 minus 1( 1113857b12X2 0(17)

or writing X2 in terms of X1

X2 g1 0( 1113857 r1 + b11 minus a1( 1113857X1 minus c1b11X1

2

b12 c1X1 minus 1( 1113857 (18)

is expression presents a discontinuity at X1 1c1 andat X2 1c2 for the f2 nullcline At this discontinuity thegrowth rate of species 1 takes the value

g1(X1 1c1) r1 minus a1c1 independently of X2 (and thesame for g2(X2 1c2)) e condition for a boundedgrowth leads to c1 le a1r1 and as in Verhulstrsquos equation thisparameter 1c1 plays the role of the carrying capacity Withthe same condition for species 2 ie c2 le a2r2 we maydefine a rectangle limited by X1 0 X1 1c1 X2 0 andX2 1c2 in whose boundary the flux vectors never pointout of the rectangle and therefore the growth is bounded

Figure 2 depicts the bounding rectangle limited by theaxis and the dashed lines 1c1 and 1c2 In Figure 2(a) theconditions c1 le a1r1 and c2 le a2r2 are fulfilled and the fluxlines are pointing inside the rectangle In Figure 2(b) theconditions are no longer satisfied but one stable solution islocated outside the rectangle allowing some flux lines to goout e asymptotic behaviour of the nullcline at X1 1c1has changed and now it rises to infinity

e intersection of both nullclines defines the stationarysolutions As the expression equation (18) is nonlinearthere can be several solutions inside the rectangle isallows more than one stable solution inside this areaseparated by saddle points As an example Figure 2 showsthe intersections of nullclines (black lines for X1 and orangelines for X2) as red points two of them are stable stationarysolutions separated by a saddle point In this example for apredator-prey system the phase space shows the typicalsolution of a stable spiral (at X1 42 and X2 79) and anew stable node at a higher population of predator and prey(at X1 200 and X2 164) Note that even though a1 doesnot fulfil the condition a1 le c1 middot r1 in this example thesystem is also bounded and stable outside the rectangleFinally the same study can be done for N species For everyspecies the value Xi 1ci can define a threshold for theinitial population for which the flux trajectories never gooutside the N-dimensional rectangle In this case free-equilibrium solutions will be harder to obtain however theJacobian at these points will have a similar expression (seeSupplementary Material)

25 Solutions with6ree Populations Ecological complexityincreases with species number Just as a little example weshow in this section how the intraspecific interaction canchange the outcomes in a 3-species predator-prey systemWe show how a positive coefficient in the intraspecific termof the prey-1 avoids the extinction Figure 3(a) shows thetime evolution of three populations two preys and onepredator the cooperation coefficient in prey-1(b11 0001) even smaller than the interspecific coefficient(b13 minus 0004) changes the initial outcome resulting in astationary population for prey-1 and predator and theextinction of prey-2

For the case of negative intraspecific interaction weshow another predator-prey system with two preys andone predator In this example the intraspecific coefficientof the predator (b33 minus 00005) allows both preys to surviveat higher populations (Figure 3(b)) the three populationsexhibit initial oscillations until they reach a stationarypopulation however the difference in the interspecificcoefficient (b13 minus 0004 and b23 minus 00045) makes the

Complexity 5

prey-1 stationary population to be higher than that ofprey-2

3 Results

Here we show the great variety of scenarios of ecologicalinteractions that this general model is capable of producinge aim of this section is to show the great richness of themodel but it is not an exhaustive study of the parametersWe show some examples of the solutions that the intra-specific interaction provides to the populations model withtwo populations Since exploring all the possible combi-nations of signs and ratios among the parameters would beunmanageable and redundant we only show some inter-esting cases For all the figures shown in this section wehave varied the parameters in the effective growth rate ribii and bij and we have set the limiting parameters a1

a2 000075 and c1 c2 0005

31Antagonism e solutions of the classical predator-preymodel are modified when intraspecific interactionscome into play In our following examples we have X1 asthe prey and X2 as the predators We only show obligatepredation since the facultative case only offers a minorchange

311 6e Effect of Cooperation among Prey e predator-prey system without any intraspecific interaction has onlytwo free-equilibrium solutions one convergent spiral andone unstable solution located at the carrying capacity ofthe prey e addition of cooperation among the pop-ulation of preys can generate a new stable solution Besides

the well known oscillatory solution we may find a newstable node at high population values separated by asaddle point Figure 4(a) shows the phase space andtrajectories (with the stationary solutions as red points)for a predator-prey system when preys (X1) exhibitpositive intraspecific interaction (cooperation) Phasetrajectories keep around the stable spiral for low pop-ulations however the saddle point defines a new basintowards the new stable solution for high population values(note that the intraspecific parameter b11 00028 islower than the absolute value of the interspecific pa-rameter b12 minus 00036)

If the detrimental interspecific interaction becomes lessharmful the original stable spiral may disappear and theonly stable solution is the coexistence at the carrying ca-pacity (Figure 4(b) with b12 minus 00036)

When the intraspecific interaction is greater than theinterspecific interactions in our example |b12| b21 lt b11 anew dynamic appears e spiral becomes unstable and thetrajectories go outwards as this stationary solution is in therepulsion basin of the saddle point the trajectories cannot goout and they will remain in a closed orbit ie in a limitcycle In Figure 5(a) (with b11 00036 and b12 minus 00072 inaddition to representing the trajectories and the stationarysolutions) we depict 3 initial points (in green yellow andorange) corresponding to the time evolution picture shownbelow e intermediate solution that appeared due to thecooperation term acts as a threshold between the spiral andthe coexistence located at the carrying capacity of the preywhich remains as a stable solution Now if we decrease theintraspecific parameter the saddle point moves towards thecarrying capacity all the stationary solutions become un-stable and all the trajectories fall into the limit cycle

200

150

100

50

0

X2

X1

0 50 100 150 200

(a)

200

150

100

50

0

X2

X1

0 50 100 150 200

(b)

Figure 2 Nullclines and phase space for an antagonistic system where both populations cooperate intraspecifically Dashed lines representXi 1ci and while solid lines are the nullclines (orange for X2 and black for X1) (a)a1 00008 fulfils the condition a1 gt c1 middot r1 and fluxlines inside the rectangle do not point out of this region (b)a1 00007lt c1 middot r1 and some flux lines go out of the rectangle Parametersr1 015 r2 minus 015 b11 00028 b12 minus 00034 b21 00072 b22 00005 a2 000075 and c1 c2 0005

6 Complexity

(Figure 5(b)) e corresponding time evolution(Figure 5(b)) shows fluctuating population for all initialpoints

312 6e Effect of Cooperation among Predators In Fig-ure 6 we show the effect of the intraspecific interactions onlyon predators As in the previous case without any intra-specific interaction the system has only two free-equilib-rium points one convergent spiral and one unstablesolution located at the carrying capacity of the prey eaddition of cooperation among predators can generate a

pair of new solutions both of them corresponding to partialextinctions of prey e effect is the same that we showedfor one population in Figure 1 but acting on the predatoraxis us cooperation among predators introduces asimilar effect of facultative predation We tested two dif-ferent values of predators cooperation parameter b22 to seeits direct influence Although in both cases the cooperativeterm is greater than predation ie b21 lt b22 we can see thatat lower values of cooperation almost no effect is notablebut at greater values two partial extinctions of prey appearone stable and one unstable a saddle-node bifurcationisallows predators to survive without preys when cooperation

0

50

100

150

200

Popu

latio

n

400 10008006002000Time

Prey-1Prey-2Predator

(a)

200 800 1000400 6000Time

0

50

100

150

200

Popu

latio

n

Prey-1Prey-2Predator

(b)

Figure 3 Population evolution in a predator-prey system with two preys (a) prey-1 with cooperation b11 0001 (r1 r2 015r3 minus 015 b13 b23 minus 0004 and b31 b32 0004) (b) predator with competition b33 minus 00005 (r1 r2 015 r3 minus 015b13 minus 0004 b23 minus 00045 and b31 b32 0001)

200

250

150

100

50

0

X2

0 25 50 75 100 125 150 175 200X1

(a)

200

250

150

100

50

0

X2

0 25 50 75 100 125 150 175 200X1

(b)

Figure 4 Phase space and trajectories for two populations involved in a predator-prey interaction To the left we have a case with a lowerb11 00028 than to the right b11 00035 Cooperation among prey allows a new intermediate solution which is unstable and acts in thesame way as in Figure 1 Also as greater cooperation decreases the predatory term and the relation may become commensalistic at somepoints Here r1 015 r2 minus 015 b12 minus 00036 and b21 00072

Complexity 7

reaches a certain limit In Figure 6(a) we have the case inwhich cooperation is weaker and in Figure 6(b) the case inwhich is mildly stronger e coexistence located at thecarrying capacity of the prey remains unstable

32Competition In the case of competition the principle ofcompetitive exclusion stands that the stable solution is thepartial extinction but if interaction parameters are weakanother feasible stable solution is a coexistence point [17]However by including intraspecific interactions the coex-istence could become stable for higher or lower values of theinterspecific interaction parameters For a range of positiveintraspecific parameters partial extinctions and the totalcarrying capacity could be stable at the same time Adding apositive intraspecific interaction term (cooperation) in one

species may induce a new saddle point defining two basinsone towards partial extinction of this species and the otherone to the system carrying capacity When cooperationoccurs in both species these two saddle points and the origindefine a central attraction basin towards the system carryingcapacity meanwhile outside this basin the system evolvestowards one species extinction as per the principle ofcompetitive exclusion (see Figure 7(a)) When we havenegative intraspecific parameters the carrying capacitybecomes unstable and the only stable solutions are thepartial extinctions however due to the intraspecific inter-action these points occur at a population below its carryingcapacity Both effects can be seen as consequences of in-traspecific cooperation and competition in the same way asfor one population in Figure 1 Cooperation induces new

150

100

50

X2

X1

0 50 100 150 200

0

200

(a)

X1

0 50 100 150 200

150

100

50

X2

0

200

(b)

0

50

100

150

200

Popu

latio

n

20 40 60 80 100t (au)

x1 [0] = 50x2 [0] = 120x1 [0] = 150

x2 [0] = 120x1 [0] = 100x2 [0] = 30

(c)

50

100

150

200Po

pula

tion

20 40 60 80 100t (au)

x1 [0] = 50x2 [0] = 120x1 [0] = 150

x2 [0] = 120x1 [0] = 100x2 [0] = 30

(d)

Figure 5 Phase space and trajectories for two populations involved in a predator-prey interaction We show here a special case where thecoexistence spiral solution diverges and become unstable When that happens a limit cycle appears To the left we have a case with a smallerpredation ie b12 minus 0006435 than to the right where b12 minus 00072 In both cases b11 b21 00036 which means that both populationsbenefit the same from population X1 but the predatory effects of X2 on X1 are stronger on the right For greater cooperation values theintermediate solution might even disappear as it is shown on the right e green blue and yellow dots in the phase space mark the initialconditions of the simulations located below Here r1 015 and r2 minus 015

8 Complexity

solutions as partial carrying capacities and intraspecificcompetition as partial extinctions

33 Mutualism e logistic-mutualistic model exhibits inaddition to the total and partial extinctions two feasiblefinite solutions (5) the larger one corresponds to the casewhere both populations reach their carrying capacities andthe lower one is a saddle point that allows us to define a

survival watershed By adding intraspecific interactions newpartial extinctions and carrying capacities could appear

331 Obligate-Obligate Mutualism For the sake of sim-plicity we only expose the case of equal sign in the pa-rameters for both species ie r1 r2 lt 0 and b12 b21 gt 0 InFigure 8 we show the phase space for two populationsinvolved in a mutual obligatory mutualism with two

200

250

150

100

50

0

X2

0 25 50 75 100 125 150 175 200X1

(a)

200

250

150

100

50

0

X2

0 25 50 75 100 125 150 175 200X1

(b)

Figure 6 Phase space and trajectories for two populations involved in an antagonist interaction To the left we have a case with a lowerb22 0004 than to the right b22 0005 Cooperation among predators allows two new partial extinctions of prey one stable and oneunstable in the same way in Figure 1 but on the predators axise coexistence located at the carrying capacity of the prey remains unstableHere r1 015 r2 minus 015 b11 0 b12 minus 00072 and b21 00036

0 25 50 75 100 125 150 175 200X1

X2

200

175

150

100

125

75

50

25

0

(a)

0 25 50 75 100 125 150 175 200X1

X2

200

175

150

100

125

75

50

25

0

(b)

Figure 7 Phase space and trajectories for two populations involved in competition with positive intraspecific interaction We used twodifferent combinations of b11 b22 to see the influence of intraspecific cooperation and competition (a) b11 b22 00019 and we have thecase in which both populations are cooperative and two new solutions appear together with a basin towards the carrying capacity of thesystem (b) b11 b22 minus 0001 and we have the case in which both are competitive Noting that when both populations are cooperativepartial carrying capacities appear and they are both unstable And when both populations are competitive partial extinctions appear insteadalthough stable and below the carrying capacities Here r1 r2 015 and b12 b21 minus 0002

Complexity 9

different values of the cooperation coefficients bii InFigure 8(a) with weak cooperation the phase space exhibitstwo free-equilibrium points the stable carrying capacity anda saddle point defining a survival watershed as in [5]However with strong intraspecific interaction (Figure 8(b))four new unstable solutions can appear two saddle pointsand two unstable fixed nodes corresponding to partialextinctions As in the case of one population (see Figure 1)the new saddle points are the thresholds Whenever apopulation is higher than this threshold it will never goextinct e total extinction basin is limited by the curve

passing through the nontrivial saddle point and these newunstable fixed nodes

On the contrary when mutualistic species exhibits neg-ative intraspecific interactions as in Figure 9 the stablecarrying capacity moves towards the saddle point(Figure 9(a)) And eventually when this negative term is highenough these two solutions collide and total extinction re-mains as the exclusive stable stationary solution (Figure 9(b))

In the case of one cooperative population and onecompetitive population the system exhibits this asymmetryagain a new saddle point in the cooperative population axissets a survival threshold Above it the system always evolves

200

150

175

100

125

50

25

75

0

X2

0 25 50 75 100 125 150 175 200X1

(a)

200

150

175

100

125

50

25

75

0

X2

0 25 50 75 100 125 150 175 200X1

(b)

Figure 8 Obligate-obligate mutualism with cooperation in two populations (a) we have the case where b11 b22 00001 which meansthat intraspecific cooperation is lower than mutualism (b) we have b11 b22 00045 which means that both intraspecific cooperation andmutualism weight the same Here r1 r2 minus 015 and b12 b21 00045

0 25 50 75 100 125 150 175 200X1

X2

200

175

150

100

125

75

50

25

0

(a)

0 25 50 75 100 125 150 175 200X1

X2

200

175

150

100

125

75

50

25

0

(b)

Figure 9 Obligate-obligate mutualism with competition in two populations (a) we have the case where b11 b22 minus 00001 which meansthat intraspecific competition is lower thanmutualism (b) we have b11 b22 minus 000062868 whichmeans that intraspecific competition hasstronger effects than mutualism Here r1 r2 minus 015 and b12 b21 0005

10 Complexity

towards the coexistence solution and will never go extinctand it is shown in Figure 10

332 Facultative-Facultative Mutualism When bothgrowth rates r1 and r2 are positive total extinction is anunstable solution and the carrying capacity is stable(Figure 11(a)) However when both populations exhibitnegative intraspecific interactions the maximum systemcarrying capacity may become unstable and a new stable

finite solution emerges at lower populations (Figure 11(b))as one expects following the one population solution withintraspecific competition (see Figure 1) In Figure 11(a) theintraspecific interaction generates four partial extinctions asunstable stationary solutions (two saddle points and twounstable nodes) In Figure 11(b) with higher negative in-traspecific interaction two extra solutions appear as partialcarrying capacities and the total carrying becomes unstableIn this case the system exhibits 9 positive stationary

200

150

100

50

0

X2

X1

0 50 100 150 200

(a)

200

150

100

50

0

X2

X1

0 50 100 150 200

(b)

Figure 10 Obligate-obligate mutualism with positive and negative intraspecific interaction (a) we have the case where b11 minus 0002 andb22 0002 which means that intraspecific competition of X1 is the same that intraspecific cooperation of X2 and both interactions weightlower than mutualism (b) we have b11 minus 00045 and b22 00045 which means that intraspecific competition of X1 weights the same thanmutualism and intraspecific cooperation of X2 Here r1 r2 minus 015 and b12 b21 00045

0 25 50 75 100 125 150 175 200X1

X2

200

175

125

150

100

75

50

25

0

(a)

0 25 50 75 100 125 150 175 200X1

X2

200

175

125

150

100

75

50

25

0

(b)

Figure 11 Facultative-facultative mutualism with negative intraspecific interaction (a) we have the case where b11 b22 minus 0002 whichmeans that intraspecific competition is weaker than mutualism (b) we have b11 b22 minus 0008 which means that intraspecific competitionis stronger than mutualism Here r1 r2 015 and b12 b21 0005

Complexity 11

solutions four saddle points four unstable points and onlyone stable solution

In the case of facultative mutualism with different in-traspecific interactions one of them is beneficial and theother one is harmful the carrying capacity could be reducedfor populations with negative intraspecific interaction whileits partner with positive intraspecific interaction will growuntil reaching its own saturation Figure 12 depicts thisscenario In Figure 12(a) competition is weaker than co-operation and the total carrying capacity is the stable sta-tionary solution In Figure 12(b) competition is strongerthan cooperation and the total carrying capacity becomesunstable As before competition only generates unstable apartial extinction while cooperation pushes the coexistencesolution into a transcritical bifurcation

4 Conclusions

In the title of the paper we ask how simple a populationdynamics model should be To address the discussion wehave introduced the intraspecific interactions in the [5]model using their same philosophy to include new termsese appear in the first term of the interaction representingthe effective growth rate and in the logistic brake to balancethe first term With respect to the previous model thismodification introduces two new terms biiX

2i and minus cibiiX

3i

regarding the intraspecific interactions Furthermore wehave generalized the model allowing the parameters thatdefine the interactions bij to be positive or negative

In our opinion the ecological reason to introduce dif-ferent intraspecific interactions is supported by observa-tions cooperative and competitive intraspecific interactionsare widely known in a wide variety of ecological systemsfrom social insects to microbial communities ey havebeen overseen by population dynamics modelling whichmainly focused on interactions with the environment or

interspecific interactions (see for example the historicalsequence developed in [18])

Furthermore the cubic term offers an interesting be-haviour from the mathematical point of view As AlAdwaniand Saavedra [11] explain that new high-order terms canintroduce new free-equilibrium solutions but it is necessarythat these solutions will be feasible and of course with aclear ecological meaning In this way several authors haveused high-order interactions to improve the stability ordiversity of ecological models For example Letten andStouffer [9] show the advantages of the high-order termsintroducing nonadditive density-dependent effects the au-thors study the influence of the high-order interactions inthe competitive communities Or Grilli et al [19] show howthe high-order interactions increase the stability of thesystems In our model the term minus cibiiX

3i introduces 2 new

free-equilibrium solutions (see Supplementary Material(available here)) that in our opinion can explain ecologicalsituations that were not well explained before with thepopulation dynamics equations

Delving into the idea of high-order interactions Bairey et al[10] introduce 3-way or 4-way terms overcoming the pairwiseinteractionsese terms are intended to simulate the effect thatinteractions between species are modulated by one or morespeciesis idea is inspiring butwe believe that simplermodelslike ours that use polynomial terms and pairwise interaction canstill explain many ecological landscapes Every time that weincrease the order of a new term it is more difficult to define itand their corresponding parameters in the field

We would like to highlight that the inclusion of theintraspecific terms shows new solutions that could representmore complex ecological landscapes For example the caseof predator-prey system with positive intraspecific term inthe preys exhibits a new solution with a steady state at largepopulations is solution could represent the way herds actas a defensive mechanism for preys [20 21] Also large herds

0 25 50 75 100 125 150 175 200X1

X2

200

175

150

100

125

75

50

25

0

(a)

0 25 50 75 100 125 150 175 200X1

X2

200

175

150

100

125

75

50

25

0

(b)

Figure 12 Facultative-facultative mutualism with intraspecific competition and cooperation (a) we have the case where b11 minus 0002 andb22 0008 (b) we have b11 minus 0008 and b22 0002 Here r1 r2 015 and b12 b21 0005

12 Complexity

of zebras or wildebeest seem to be stable in time in [22] theauthors presented data of the Kruger National Park in SouthAfrica that showed a stable and increasing population ofzebras and wildebeest (more than 10000 individuals) over aperiod of twenty years with more or less stable population oflions (around 400 individuals) Or the effects of intraspecificcompetition can act as a regulatory mechanism Polis [23]showed that intraspecific predation acts in a reinforced wayhigher populations decrease the resources available for in-dividuals reducing their growing rates and promotingsmaller and weaker individuals those are more easily killedor eaten which increases the per capita food level both byreducing the population and by satiating the cannibalists

e main advantage of this general model (equation (3))is that it can be used to describe any ecological regime andthat it carries its own saturation mechanism that avoids theldquoorgy of mutual benefactionrdquo of [24] Stucchi [25] showedusing a simplified generalized model studying a nurserypollination system and modelling all the interspecific in-teractions with the same functional is allowed a clearinterpretation of the parameters of the whole system and anunambiguous way to compare them Furthermore Stucchi tal [26] showed that intraspecific interactions in a predator-prey system might lead to diffusion-driven instabilities

Finally we would like to venture to discuss some morespeculative ideas Nowadays there are some attempts to modeltransitions from antagonistic to mutualistic interspecific rela-tionships limited by the fact that they deal with differentmathematical functionals for mutualism and antagonism[27ndash29] ese models include changes that arise continuallyfromone regime to another but treating the transition only in adescriptive way In addition adaptive changes are modelledthrough parameter changing systems where parameters havetheir own dynamic equations but these models are still limitedto specific ecological regimes either antagonistic or mutualistic[30ndash32] However if one may adequately define the dynamicsof the parameters in a general model of ecological interactionsit may reflect a deeper view of nature where ecology meetsevolution us by including evolutionary changes in ourmodel one may be capable of modelling transitions due tomutations and natural selection which is surely the way howtransitions on ecological regimes occur in nature

Data Availability

is is a theoretical study and we do not have experimentaldata

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to the study

Acknowledgments

is work was supported by the Ministry of EducationCulture and Sport of Spain (PGC2018-093854-B-100)

Supplementary Materials

Quantifying the effect of the intraspecific terme Jacobianmatrix for N species (Supplementary Materials)

References

[1] A J Lotka Elements of Physical Biology Williams andWilkinsCompany Baltimore MD USA 1925

[2] V Volterra ldquoFluctuations in the abundance of a speciesconsidered mathematicallyrdquo Nature vol 118 no 2972pp 558ndash560 1926

[3] J Bascompte ldquoDisentangling the web of liferdquo Science vol 325no 5939 pp 416ndash419 2009

[4] J Bascompte ldquoMutualistic networksrdquo Frontiers in Ecologyand the Environment vol 7 no 8 pp 429ndash436 2009

[5] J Garcıa-Algarra J Galeano J M Pastor J M Iriondo andJ J Ramasco ldquoRethinking the logistic approach for pop-ulation dynamics of mutualistic interactionsrdquo Journal of6eoretical Biology vol 363 pp 332ndash343 2014

[6] A M Dean ldquoA simple model of mutualismrdquo 6e AmericanNaturalist vol 121 no 3 pp 409ndash417 1983

[7] D H Wright ldquoA simple stable model of mutualism incor-porating handling timerdquo 6e American Naturalist vol 134no 4 pp 664ndash667 1989

[8] U Bastolla M A Fortuna A Pascual-Garcıa A FerreraB Luque and J Bascompte ldquoe architecture of mutualisticnetworks minimizes competition and increases biodiversityrdquoNature vol 458 no 7241 pp 1018ndash1020 2009

[9] A D Letten and D B Stouffer ldquoe mechanistic basis forhigher-order interactions and non-additivity in competitivecommunitiesrdquo Ecology Letters vol 22 no 3 pp 423ndash436 2019

[10] E Bairey E D Kelsic and R Kishony ldquoHigh-order speciesinteractions shape ecosystem diversityrdquo Nature Communi-cations vol 7 no 1 pp 1ndash37 2016

[11] M AlAdwani and S Saavedra ldquoIs the addition of higher-orderinteractions in ecological models increasing the under-standing of ecological dynamicsrdquo Mathematical Biosciencesvol 315 Article ID 108222 2019

[12] P F Verhulst ldquoNotice sur la loi que la population suit dansson accroissementrdquo Correspondance Mathematique et Phy-sique vol 10 pp 113ndash117 1838

[13] E O Wilson 6e Insect Societies Belknap Press CambridgeMA USA 1971

[14] B Stadler and A F G Dixon ldquoEcology and evolution ofaphid-ant interactionsrdquo Annual Review of Ecology Evolutionand Systematics vol 36 no 1 pp 345ndash372 2005

[15] M Stucchi and J Figueroa ldquoLa avifauna de las islas Lobos deAfuera y algunos alcances sobre su biodiversidadrdquo Asocia-cion Ucumari Lima Peru Reporte de Investigacion N 22006

[16] W C Allee Animal Aggregations A Study in General Soci-ology University of Chicago Press Chicago IL USA 1931

[17] J D Murray Mathematical Biology I An IntroductionSpringer Berlin Germany 1993

[18] P Turchin Complex Population Dynamics A 6eoreticalempirical Synthesis (MPB-35) Princeton University PressPrinceton NJ USA 2003

[19] J Grilli G Barabas M J Michalska-Smith and S AllesinaldquoHigher-order interactions stabilize dynamics in competitivenetwork modelsrdquoNature vol 548 no 7666 pp 210ndash213 2017

[20] D I Rubenstein On Predation Competition and the Ad-vantages of Group Living 205ndash231 Springer US Boston MAUSA 1978

Complexity 13

[21] J Berger ldquoldquoPredator harassmentrdquo as a defensive strategy inungulatesrdquo American Midland Naturalist vol 102 no 1pp 197ndash199 1979

[22] T H Fay and C Greeff ldquoLion wildebeest and zebra apredator-prey modelrdquo Ecological Modelling vol 196 no 1-2pp 237ndash244 2006

[23] G A Polis ldquoe evolution and dynamics of intraspecificpredationrdquo Annual Review of Ecology and Systematics vol 12no 1 pp 225ndash251 1981

[24] R M May ldquoModels for two interacting populationsrdquo in6eoretical Ecology Principles and Applications pp 78ndash104Oxford University Press Oxford UK 1981

[25] L Stucchi L Gimenez-Benavides and J Galeano ldquoe role ofparasitoids in a nursery-pollinator system a population dy-namics modelrdquo Ecological Modelling vol 396 pp 50ndash582019

[26] L Stucchi J Galeano and D A Vasquez ldquoPattern formationinduced by intraspecific interactions in a predator-prey sys-temrdquo Physical Review E vol 100 no 8 2019

[27] V I Yukalov E P Yukalova and D Sornette ldquoModelingsymbiosis by interactions through species carrying capac-itiesrdquo Physica D Nonlinear Phenomena vol 241 no 15pp 1270ndash1289 2012

[28] C Neuhauser and J E Fargione ldquoA mutualism-parasitismcontinuum model and its application to plant-mycorrhizaeinteractionsrdquo Ecological Modelling vol 177 no 3-4pp 337ndash352 2004

[29] S Kefi V Miele E A Wieters S A Navarrete andE L Berlow ldquoHow structured is the entangled bank esurprisingly simple organization of multiplex ecologicalnetworks leads to increased persistence and resiliencerdquo PLoSBiology vol 14 no 8 Article ID e1002527 2016

[30] U Dieckmann and R Law ldquoe dynamical theory of co-evolution a derivation from stochastic ecological processesrdquoJournal of Mathematical Biology vol 34 no 5-6 pp 579ndash6121996

[31] A White and R G Bowers ldquoAdaptive dynamics of Lotka-Volterra systems with trade-offs the role of interspecificparameter dependence in branchingrdquo Mathematical Biosci-ences vol 193 no 1 pp 101ndash117 2005

[32] J N Holland D L DeAngelis and S T Schultz ldquoEvolu-tionary stability of mutualism interspecific population reg-ulation as an evolutionarily stable strategyrdquo Proceedings of theRoyal Society B Biological Sciences vol 271 pp 1807ndash18142004

14 Complexity