# The “Destabilizing” Effect of Cannibalism in a Spatially Explicit Three-Species Age Structured Predator-Prey Model

Authors: Aladeen Al Basheer; Jingjing Lyu; Adom Giffin; Rana D. Parshad
Journal: Complexity
 (2017)

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## Abstract

Cannibalism, the act of killing and consumption of conspecifics, is generally considered to be a stabilising process in ODE models of predator-prey systems. On the other hand, Sun et al. were the first to show that cannibalism can cause Turing instability, in the classical Rosenzweig-McArthur two-species PDE model, which is an impossibility without cannibalism. Magnússon’s classic work is the first to show that cannibalism in a structured three-species predator-prey ODE model can actually be destabilising. In the current manuscript we consider the PDE form of the three-species model proposed in Magnússon’s classic work. We prove that, in the absence of cannibalism, Turing instability is an impossibility in this model, forany range of parameters. However, theinclusion of cannibalism can cause Turing instability. Thus, to the best of our knowledge, we report thefirst cannibalism induced Turing instability result, in spatially explicit three-species age structured predator-prey systems. We also show that, in the classical ODE model proposed by Magnússon, cannibalism can act as a life boat mechanism,for the prey.


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## Body

## 1. Introduction

Background. Cannibalism is defined as the act of killing and at least partial consumption of conspecifics. It is ubiquitous in natural predator-prey communities [1] and is observed in more than 1300 species in nature [2]. It occurs as a socioanthropological as well as ecological phenomenon [3, 4], in many varied forms. For example, a sect of ascetics in Northern India, called “Aghoris,” consume corpses in the belief that such ritualistic practice, orendocannibalism, will enable them to attain immortality [5]. Cannibalism is also rampant amongst certain insects. Many arachnid species practicesexual cannibalism, where the female cannibalises her mate, before, during, or after copulation. In population models, the inclusion of cannibalism can bring about interesting and sometimes counter intuitive dynamics, such as thelife boat mechanism. Therein the act of cannibalism causes persistence in a population, otherwise doomed to go extinct [6]. The effects of climate change induced cannibalism have also been intensely investigated recently [7–9].A detailed survey of the current and past mathematical literature on cannibalism shows that various types of predator-prey models have been investigated [6, 10–16]. Most works in the literature consider cannibalism in the predator species. However, there are a few works that involve cannibalism among prey as well [17]. In most models, cannibalism is seen to have a strong stabilising influence [11, 14]. But some works do report that it can destabilize as well [18]. Since species disperse in space in search of food, shelter, and mates and to avoid predators, spatially dispersing populations are often modeled via partial differential equations (PDE)/spatially explicit models of interacting species [19–22]. Of particular interest to many is the phenomenon of Turing instability, first introduced by Turing [23], which shows that diffusion in a system can lead to pattern forming instabilities. In the context of cannibalism, there are very few works in the literature that consider spatially explicit models and investigate Turing instability therein. Sun et al. [24] were the first to show that cannibalism in the classical diffusive Rosenzweig-McArthur model can bring about Turing patterns. This is an impossibility in the system without cannibalism. Fasani and Rinaldi [25] generalised the cannibalism term from [24] and found that cannibalism can still bring about Turing patterns. The above models consider cannibalism amongst the predators. In [17] we found that cannibalism amongst prey can also bring about Turing patterns, in the spatially explicit Holling-Tanner model, with ratio dependent functional response.One approach in the literature, when considering ODE models, has been to structure the predator class into adults and juveniles and allow cannibalism of the juvenile predator by the adult predator. This essentially yields a three-species age structured ODE model. The reader is referred to [12, 14, 27], and again therein cannibalism is seen to provide a stabilising influence, depending on parameter regime. However, depending on model structure and parameter regimes, cannibalism can also destabilize [1, 17, 18]. To the best of our knowledge, there is acomplete lack of investigation of the effects of cannibalism, on the stability of spatially explicit/PDE three-species predator-prey models, with age structure (or without).Classical Model of Magnússon. To this end we investigate the spatially explicit version of the model first proposed in [18] by Magnússon and further analyzed by Kaewmanee et al. [26, 28]. Magnússon proposed and investigated the following structured three-species predator-prey model, where X(t), Y(t), and Z(t) are the biomass of an adult predator, a juvenile predator, and prey, at a given time t. The interaction between these three classes is described below:(1)dXdt=-M1X+AY+CXZ+γ′SXYdYdt=RX-AY-M2Y-SXYdZdt=TZ-UZ2-VXZ.Essentially the predator species is structured into adultsX(t) and juveniles Y(t), with the adult predators feeding on a prey species Z(t). Note all parameters are positive constants, and the system goes along with suitable positive initial conditions. M1 is the death rate of adult predators, M2 is the death rate of juvenile predators, A is the growth rate of adults predators (juveniles grow into adults), R is the growth rate of juveniles due to adults, C is the predators growth rate due to preying, T is the birth rate of prey, and V is the predators attack rate on prey. S is the cannibalism rate, and γ′ is the conversion efficiency of eaten juveniles into adults. For further details see [18]. Magnússon finds that, in this particular model, the effect of including cannibalism is todestabilise the interior equilibrium, leading to limit cycle dynamics, through a Hopf bifurcation. This result iscontrary to many others in the literature, where cannibalism is found to be stabilising. Note one of the problematic issues in [18] is the scaling of the predator attack rate parameter, V, which scales like V≈1/S, where S is the cannibalism rate parameter. A possible reasoning here is that if V≈1/S, then S≈1/V. So if predation/attack rate on regular prey is high, or V≫1, there is almost no need for cannibalism, and S≪1. A closer examination of the model reveals two causes for concern:(i)
This scaling does not permit one to consider the limiting case of no cannibalism (S=0), in a tractable way.(ii)
There is a prey free equilibrium or no “cost” to the adult as he cannibalises the juvenile.Rescaled Model of Tang. Part of these issues was first addressed in [26], where a more refined form of scaling is introduced.(2)u1=VM1X,v1=VAM12Y,r1=CM1Z,τ=M1Ct,where t1=T/M1, m=A+M2/M1, r=RA/M12, u=U/C, γ=γ′M1/A, s=S/V.This yields the following nondimensionalised system, with fewer parameters(3)du1dτ=-u1+v1+u1r1+γsu1v1dv1dτ=ru1-mv1-su1v1dr1dτ=t1r1-ur12-u1r1.Here typicallyγ<1, but not necessarily. Now the cannibalism parameter s can be followed closely. In fact one can tractably take the limit as s→0, to obtain the model without cannibalism,(4)du1dτ=-u1+v1+u1r1dv1dτ=ru1-mv1dr1dτ=t1r1-ur12-u1r1.The results that Kaewmanee and Tang find are in accordance with Magnússon; that is increasing the cannibalism parameter can lead to the stable interior equilibrium losing stability, leading to limit cycle dynamics. Kaewmanee and Tang’s results, unlike Magnússon,are not limited to the case where there is high juvenile mortality and/or low adult recruitment. However, the Hopf bifurcation analysis as worked out in [26] isincorrect.Our primary contributions in the current manuscript are the following:(i)
We explore the spatially explicit forms of the models, (3) and (4), and show global existence of solutions via Theorem 2 and Corollary 3.(ii)
We show that the spatially explicit model without cannibalismdoes not possess Turing instability inany parameter regime via Theorem 4. However,introducing cannibalism cancertainly bring about Turing instability, via Theorem 14. To the best of our knowledge, this is thefirst cannibalism induced Turing instability result, in age structured three-species PDE models.(iii)
We derive the correct form of Hopf bifurcation in the classic ODE model proposed in [18] via Theorem 15. Note again that the previous Hopf bifurcation analysis as worked out in [26] isincorrect.(iv)
We numerically show that cannibalism can act as a life boat mechanism inthe prey and conjecture that it can act as a life boat mechanism in both predator and prey via Conjectures 16 and 18.(v)
We discuss ecological interpretations, based on our work.We relegate the details of the ODE stability analysis, as well as generic Turing instability conditions, to the Appendix. Some of these are also readily available in the literature [19, 26], so they are included in the Appendix only for completeness and benefit of the reader.


## 2. Spatially Explicit Model

Since species disperse in space in search of food, shelter, and mates and to avoid predators, spatially dispersing populations are often modeled via partial differential equations (PDE)/spatially explicit models of interacting species [19–22]. In this section we shall investigate the effects of diffusion induced instability in the models (4) and (3). We state the form of the one-dimensional spatially explicit model with no cannibalism,(5)ut1=d1uxx1-u1+v1+u1r1vt1=d2vxx1+ru1-mv1rt1=d3rxx1+t1r1-ur12-u1r1,γ<1, and our domain Ω=[0,π].We prescribe Neumann boundary conditions(6)ux1=vx1=rx1=0and suitable positive initial conditions(7)u1x,0>0,v1x,0>0,r1x,0>0.Hereu1(x,·), v1(x,·), r1(x,·) are the concentrations/population densities of the prey, the juvenile predator, and the adult predator, at any given time t, respectively. We next state the form of the one-dimensional spatially explicit model with cannibalism(8)ut1=d1uxx1-u1+v1+u1r1+γsu1v1(9)vt1=d2vxx1+ru1-mv1-su1v1(10)rt1=d3rxx1+t1r1-ur12-u1r1,γ<1, and our domain Ω=[0,π].We prescribe Neumann boundary conditions(11)ux1=vx1=rx1=0and suitable positive initial conditions(12)u1x,0>0,v1x,0>0,r1x,0>0.Hereu1(x,·), v1(x,·), r1(x,·) are the concentrations/population densities of the prey, the juvenile predator, and the adult predator, at any given time t, respectively.


### 2.1. Functional Preliminaries and Local Solutions

We now present various function space notations and definitions that will be used frequently. The usual norms in the spacesLp(Ω), L∞(Ω), and CΩ¯ are, respectively, denoted by(13)upp=1Ω∫Ωuxpdx,u∞=maxx∈Ω⁡ux.It is well known that, under the “regularizing effect principle” to prove global existence of solutions to (8)–(10), [29], it suffices to derive uniform estimates on Lp, norms of the reaction terms, on 0,Tmax, for some p>n/2, where n is the spatial dimension of the domain Ω, and the reaction terms in our setting are given in (8)–(10). Here Tmax denotes the eventual blowing-up time in L∞(Ω). The following local existence result is well known [30].Lemma 1.
System (8)–(10) admits a unique, classical solution (u1,v1,r1) on [0,Tmax[×Ω. If Tmax<∞ then(14)limt↗Tmax⁡u1t,·∞+v1t,·∞+r1t,·∞=∞.Proof.
Since the reaction terms are continuously differentiable in the positive octant, then for any initial data inCΩ¯ or Lp(Ω), p∈1,+∞, it is easy to check directly their Lipschitz continuity on bounded subsets of the domain of a fractional power of the operator I3d1,d2,d3tΔ, where I3 is the three-dimensional identity matrix, Δ is the Laplacian operator, and t denotes the transposition.


### 2.2. Control of Mass andL1(Ω) Estimates

Note, in order to derive global existence via application of [29], we are required to derive uniform in time Lp estimates on the reaction terms in (8)–(10), for p>1/2. We will restrict ourselves to the case n=1. To this end (8) poses a difficulty via the γsu1v1 term. We proceed by multiplying (9) by γ and adding up equations (8) and (9) to yield(15)ut1+γvt1=d1uxx1+d2γvxx1+rγ-1u1+1-γmv1+u1r1.We now define the grouped variableW=u1+γv1 and integrate the above equation in space to obtain(16)ddt∫ΩWx,tdx=∫Ωrγ-1u1+1-γmv1+u1r1dx≤max⁡rγ-1,1-γm,r1∞∫Ωu1+v1dx≤Cmax⁡rγ-1,1-γm,r1∞∫Ωu1+γv1dx=Cmax⁡rγ-1,1-γm,r1∞∫ΩWx,tdx.Now the uniform in timeL1 bound on W and thus on u1,v1 easily follows via applying Gronwall’s lemma on (16), on any time interval [0,T], to yield(17)u1L1Ω≤u01+γv01L1ΩeCmax⁡rγ-1,1-γm,r1∞T,(18)v1L1Ω≤u01+γv01L1Ω1γeCmax⁡rγ-1,1-γm,r1∞Twith the bound depending on only the final time t=T. This follows as the equation for r1 is amenable to make essentially any estimate on r1, by comparing it to the diffusive logistic equation. We now deal with the problematic term which is the u1v1 term. However, integrating (9) in Ω×[0,T]=ΩT we obtain(19)∫Ωv1·,Tdx+s∫ΩTu1v1dxdt=r∫ΩTu1-m∫ΩTrv1+∫Ωv01dx.Thus(20)u1v1ΩT≤rs∫ΩTu1+1s∫Ωv01dx≤rs∫0Tu01+γv01L1ΩeCmax⁡rγ-1,1-γm,r1∞tdt+1sv01L1Ω=rsu01+γv01L1ΩeCmax⁡rγ-1,1-γm,r1∞TCmax⁡rγ-1,1-γm,r1∞-1u01+γv01L1Ω.This follows from theL1(Ω) estimate on u1, via (17). Thus we have L1(ΩT) control of the problematic nonlinearity. Thus we see we can uniformly bound all of the reaction terms in L1(ΩT) and by bootstrap argument then bound them uniformly in Lp to deduce global existence [29]. So we can state the following theorem.Theorem 2.
Consider the spatially explicit three-species model with cannibalism (8)–(10), where the spatial dimension of the physical domain is n=1. Solutions to this model are classical, that is, (u1,v1,r1)∈C1(0,T;C2(Ω)), and exist globally in time.A simple corollary to Theorem2 can be stated concerning the model without cannibalism.Corollary 3.
Consider the spatially explicit three-species model without cannibalism (5). Solutions to this model are classical, that is, (u1,v1,r1)∈C1(0,T;C2(Ω)), and exist globally in time.


### 2.3. Nonexistence of Turing Instability in the Model without Cannibalism

We state and prove the following theorem.Theorem 4.
Consider the three-species spatially explicit model without cannibalism (5). The spatially homogenous steady state (u∗1,v∗1,r∗1) of this model cannot be driven unstable by diffusion, for any parameter regime.Proof.
We show this directly by checking the necessary conditions for Turing instability from [19]. Note for Turing instability to occur we have certain necessary conditions on the coefficients of the characteristic polynomial, gotten by linearizing (5), around the spatially homogenous equilibrium (u∗1,v∗1,r∗1). See the Appendix and Theorem C.1 therein. Note that the Jacobian matrix of the reaction terms of the model without cannibalism at the spatially homogenous equilibrium (u∗1,v∗1,r∗1) is given by(21)J=-rm1-u+urm+t1r-m0-1+rm0-1mum-r.
We now check the necessary conditions for Turing instability from Table1, where we take the Jii entries, from the above matrix.
We first check the coefficientc of A3,(22)c=--d1d2ur∗1-d1d3m-d2d3rm>0.We next check the coefficient c of A1A2-A3,(23)c=rmd2+d32d1+d2+d3+md1+d3d1+2d2+d3+ur∗1d2+d1d1+d2+2d3>0.We next check the coefficient d of A3,(24)d=d1mur∗1+d2r∗1-1-ur∗1+u∗1r∗1+d3-mr∗1-1-r=d1mur∗1+d2r∗1-1-ur∗1+u∗1r∗1>0.
We next check the coefficientd of A1A2-A3,(25)d=d12rm2um-r+r+2um-r+1mum-r2+m2+u∗1r∗1+d22rm2um-r+r+2um-r+1mum-r2+rm2+d32rm2um-r+2r+2um-r+m2+rm2+u∗1r∗1>0.Since all of these coefficients are positive, the result follows as an application of Theorem C.1 from the Appendix and simple contraposition. This proves the theorem.Table 1
Coefficients of cubic functionsA1(k2)A2(k2)-A3(k2) used in determining conditions for Turing instability.
Coefficient A 3 ( k 2 ) [ A 1 A 2 - A 3 ] ( k 2 ) h J 11 J 32 J 23 + J 12 J 21 J 33 - J 11 J 22 J 33 J 11 J 22 J 33 - ( J 11 + J 22 + J 33 ) ( J 11 J 22 - J 12 J 21 + J 11 J 33 + J 22 J 33 - J 23 J 32 )  - J 11 J 23 J 32 - J 12 J 21 J 33 d d 1 ( J 22 J 33 - J 32 J 23 ) + d 2 J 11 J 33 + d 3 ( J 11 J 22 - J 12 J 21 ) d 1 2 J 11 J 33 + 2 J 11 J 22 + 2 J 22 J 33 + J 33 J 33 + J 22 J 22 - J 12 J 21 + d 2 2 J 22 J 11 + 2 J 22 J 33 + 2 J 33 J 11 + J 11 J 11 + J 33 J 33 - J 21 J 12 - J 23 J 32 + d 3 2 J 22 J 11 + 2 J 22 J 33 + 2 J 33 J 11 + J 11 J 11 + J 22 J 22 - J 23 J 32 c - d 1 d 2 J 33 - d 1 d 3 J 22 - d 2 d 3 J 11 - J 11 ( d 2 + d 3 ) ( 2 d 1 + d 2 + d 3 ) - J 22 ( d 1 + d 3 ) ( d 1 + 2 d 2 + d 3 ) - J 33 ( d 1 + d 2 ) ( d 1 + d 2 + 2 d 3 ) b d 1 d 2 d 3 ( d 2 + d 3 ) ( d 1 d 1 + d 2 d 3 + d 1 d 2 + d 1 d 3 )Remark 5.
Note that Theorem4 shows that there does not exist Turing instability in the model without cannibalism (5), forany parameter regime.


### 2.4. Existence of Turing Instability with Cannibalism

Given that there is no Turing instability in the model without cannibalism (5), we now investigate the effect of including cannibalism, on Turing instability in the model. To this end we investigate the spatially explicit model (8)–(10). We first introduce several structure theorems from linear algebra in the literature [31–33].Definition 6 (strongly stable).
A real matrixA is said to be strongly stable if A-D is stable for all diagonal matrices D.Definition 7.
LetP denote the class of matrices whose signed principal minors are all positive and P0+ the class whose signed principal minors are all nonnegative, with one of each order positive.Theorem 8.
A3×3 real matrix A is strongly stable if and only if A∈P0+, and A is stable.Theorem 9.
A is strongly stable if and only if A is s-stable.Theorem 10.
If the kinetic system(26)dudt=fu,v,rdvdt=gu,v,rdrdt=hu,v,ris s-stable, then no Turing bifurcation is possible from the uniform steady state solution.Theorem 11.
If the kinetic system(27)dudt=fu,v,rdvdt=gu,v,rdrdt=hu,v,rcontains an unstable subsystem, then Turing bifurcation is possible from the uniform steady state solution.Remark 12.
We will now use the above theory to show that cannibalism can in fact induce Turing instability. We will first show that the Jacobian matrix of the reaction terms of the model with cannibalism (8)–(10), given by (A.2), isnot strongly stable, for certain values of the cannibalism parameter s. This will yield the result as a simple corollary to Theorems 9 and 10. Note similar ideas were used in [32], to show that a certain SEIR type epidemics model does not possess Turing instability, as long as the disease-caused death rate parameter is zero, but can have Turing patterns for positive disease-caused death rate.
Recall the Jacobian matrix of the reaction terms of the model with cannibalism is given by(28)J=r∗1-1+γsv∗11+γsu∗1u∗1r-sv∗1-m-su∗10-r∗10t1-2ur∗1-u∗1=free+++-0-0-.Remark 13.
Notice we write the sign ofJ11 as free, because due to the cannibalism parameter s the sign of J11=r∗1-1+γsv∗1 could now be positive or negative.Theorem 14.
Consider the three-species model with cannibalism (8)–(10) and its Jacobian matrix J of reaction terms, given by (28). Consider a parameter set for which J is stable, whilst the cannibalism parameter s satisfies(29)r∗1-1+γsv∗1>0,r∗1-1+γsv∗1-m-su∗1>0,orr∗1-1+γsv∗1+t1-2ur∗1-u∗1>0.
Then there exist diffusion coefficients(d1,d2,d3), such that the spatially homogenous steady state (u∗1,v∗1,r∗1) of (8)–(10) can be driven unstable due to Turing instability.The proof relies on extracting an unstable submatrix from (28). This is equivalent to some signed principal minor not belonging to P0+ and hence J given by (28) not being strongly stable.Proof.
Note if we look at the minor gotten by removing row 3 and column 3, we obtain the following submatrix:(30)r∗1-1+γsv∗11+γsu∗1r-sv∗1-m-su∗1.
This is clearly seen to be unstable ifr∗1-1+γsv∗1-m-su∗1>0. Also, if we look at the minor gotten by removing row 2 and column 2, we obtain the following submatrix:(31)r∗1-1+γsv∗1u∗1-r∗1t1-2ur∗1-u∗1.
This is clearly seen to be unstable ifr∗1-1+γsv∗1+t1-2ur∗1-u∗1>0.
Thus if either of these conditions are met,J given by (28) contains unstable submatrices. That is, certain signed principal minors are not in P0+. Thus via Theorem 8  J is not strongly stable, which implies via Theorem 9 that J is not s-stable. This then implies that there exist diffusion coefficients (d1,d2,d3) for which Turing instability is acertainty, via an application of Theorems 10 and 11. This proves the theorem.


## 2.1. Functional Preliminaries and Local Solutions

We now present various function space notations and definitions that will be used frequently. The usual norms in the spacesLp(Ω), L∞(Ω), and CΩ¯ are, respectively, denoted by(13)upp=1Ω∫Ωuxpdx,u∞=maxx∈Ω⁡ux.It is well known that, under the “regularizing effect principle” to prove global existence of solutions to (8)–(10), [29], it suffices to derive uniform estimates on Lp, norms of the reaction terms, on 0,Tmax, for some p>n/2, where n is the spatial dimension of the domain Ω, and the reaction terms in our setting are given in (8)–(10). Here Tmax denotes the eventual blowing-up time in L∞(Ω). The following local existence result is well known [30].Lemma 1.
System (8)–(10) admits a unique, classical solution (u1,v1,r1) on [0,Tmax[×Ω. If Tmax<∞ then(14)limt↗Tmax⁡u1t,·∞+v1t,·∞+r1t,·∞=∞.Proof.
Since the reaction terms are continuously differentiable in the positive octant, then for any initial data inCΩ¯ or Lp(Ω), p∈1,+∞, it is easy to check directly their Lipschitz continuity on bounded subsets of the domain of a fractional power of the operator I3d1,d2,d3tΔ, where I3 is the three-dimensional identity matrix, Δ is the Laplacian operator, and t denotes the transposition.


## 2.2. Control of Mass andL1(Ω) Estimates

Note, in order to derive global existence via application of [29], we are required to derive uniform in time Lp estimates on the reaction terms in (8)–(10), for p>1/2. We will restrict ourselves to the case n=1. To this end (8) poses a difficulty via the γsu1v1 term. We proceed by multiplying (9) by γ and adding up equations (8) and (9) to yield(15)ut1+γvt1=d1uxx1+d2γvxx1+rγ-1u1+1-γmv1+u1r1.We now define the grouped variableW=u1+γv1 and integrate the above equation in space to obtain(16)ddt∫ΩWx,tdx=∫Ωrγ-1u1+1-γmv1+u1r1dx≤max⁡rγ-1,1-γm,r1∞∫Ωu1+v1dx≤Cmax⁡rγ-1,1-γm,r1∞∫Ωu1+γv1dx=Cmax⁡rγ-1,1-γm,r1∞∫ΩWx,tdx.Now the uniform in timeL1 bound on W and thus on u1,v1 easily follows via applying Gronwall’s lemma on (16), on any time interval [0,T], to yield(17)u1L1Ω≤u01+γv01L1ΩeCmax⁡rγ-1,1-γm,r1∞T,(18)v1L1Ω≤u01+γv01L1Ω1γeCmax⁡rγ-1,1-γm,r1∞Twith the bound depending on only the final time t=T. This follows as the equation for r1 is amenable to make essentially any estimate on r1, by comparing it to the diffusive logistic equation. We now deal with the problematic term which is the u1v1 term. However, integrating (9) in Ω×[0,T]=ΩT we obtain(19)∫Ωv1·,Tdx+s∫ΩTu1v1dxdt=r∫ΩTu1-m∫ΩTrv1+∫Ωv01dx.Thus(20)u1v1ΩT≤rs∫ΩTu1+1s∫Ωv01dx≤rs∫0Tu01+γv01L1ΩeCmax⁡rγ-1,1-γm,r1∞tdt+1sv01L1Ω=rsu01+γv01L1ΩeCmax⁡rγ-1,1-γm,r1∞TCmax⁡rγ-1,1-γm,r1∞-1u01+γv01L1Ω.This follows from theL1(Ω) estimate on u1, via (17). Thus we have L1(ΩT) control of the problematic nonlinearity. Thus we see we can uniformly bound all of the reaction terms in L1(ΩT) and by bootstrap argument then bound them uniformly in Lp to deduce global existence [29]. So we can state the following theorem.Theorem 2.
Consider the spatially explicit three-species model with cannibalism (8)–(10), where the spatial dimension of the physical domain is n=1. Solutions to this model are classical, that is, (u1,v1,r1)∈C1(0,T;C2(Ω)), and exist globally in time.A simple corollary to Theorem2 can be stated concerning the model without cannibalism.Corollary 3.
Consider the spatially explicit three-species model without cannibalism (5). Solutions to this model are classical, that is, (u1,v1,r1)∈C1(0,T;C2(Ω)), and exist globally in time.


## 2.3. Nonexistence of Turing Instability in the Model without Cannibalism

We state and prove the following theorem.Theorem 4.
Consider the three-species spatially explicit model without cannibalism (5). The spatially homogenous steady state (u∗1,v∗1,r∗1) of this model cannot be driven unstable by diffusion, for any parameter regime.Proof.
We show this directly by checking the necessary conditions for Turing instability from [19]. Note for Turing instability to occur we have certain necessary conditions on the coefficients of the characteristic polynomial, gotten by linearizing (5), around the spatially homogenous equilibrium (u∗1,v∗1,r∗1). See the Appendix and Theorem C.1 therein. Note that the Jacobian matrix of the reaction terms of the model without cannibalism at the spatially homogenous equilibrium (u∗1,v∗1,r∗1) is given by(21)J=-rm1-u+urm+t1r-m0-1+rm0-1mum-r.
We now check the necessary conditions for Turing instability from Table1, where we take the Jii entries, from the above matrix.
We first check the coefficientc of A3,(22)c=--d1d2ur∗1-d1d3m-d2d3rm>0.We next check the coefficient c of A1A2-A3,(23)c=rmd2+d32d1+d2+d3+md1+d3d1+2d2+d3+ur∗1d2+d1d1+d2+2d3>0.We next check the coefficient d of A3,(24)d=d1mur∗1+d2r∗1-1-ur∗1+u∗1r∗1+d3-mr∗1-1-r=d1mur∗1+d2r∗1-1-ur∗1+u∗1r∗1>0.
We next check the coefficientd of A1A2-A3,(25)d=d12rm2um-r+r+2um-r+1mum-r2+m2+u∗1r∗1+d22rm2um-r+r+2um-r+1mum-r2+rm2+d32rm2um-r+2r+2um-r+m2+rm2+u∗1r∗1>0.Since all of these coefficients are positive, the result follows as an application of Theorem C.1 from the Appendix and simple contraposition. This proves the theorem.Table 1
Coefficients of cubic functionsA1(k2)A2(k2)-A3(k2) used in determining conditions for Turing instability.
Coefficient A 3 ( k 2 ) [ A 1 A 2 - A 3 ] ( k 2 ) h J 11 J 32 J 23 + J 12 J 21 J 33 - J 11 J 22 J 33 J 11 J 22 J 33 - ( J 11 + J 22 + J 33 ) ( J 11 J 22 - J 12 J 21 + J 11 J 33 + J 22 J 33 - J 23 J 32 )  - J 11 J 23 J 32 - J 12 J 21 J 33 d d 1 ( J 22 J 33 - J 32 J 23 ) + d 2 J 11 J 33 + d 3 ( J 11 J 22 - J 12 J 21 ) d 1 2 J 11 J 33 + 2 J 11 J 22 + 2 J 22 J 33 + J 33 J 33 + J 22 J 22 - J 12 J 21 + d 2 2 J 22 J 11 + 2 J 22 J 33 + 2 J 33 J 11 + J 11 J 11 + J 33 J 33 - J 21 J 12 - J 23 J 32 + d 3 2 J 22 J 11 + 2 J 22 J 33 + 2 J 33 J 11 + J 11 J 11 + J 22 J 22 - J 23 J 32 c - d 1 d 2 J 33 - d 1 d 3 J 22 - d 2 d 3 J 11 - J 11 ( d 2 + d 3 ) ( 2 d 1 + d 2 + d 3 ) - J 22 ( d 1 + d 3 ) ( d 1 + 2 d 2 + d 3 ) - J 33 ( d 1 + d 2 ) ( d 1 + d 2 + 2 d 3 ) b d 1 d 2 d 3 ( d 2 + d 3 ) ( d 1 d 1 + d 2 d 3 + d 1 d 2 + d 1 d 3 )Remark 5.
Note that Theorem4 shows that there does not exist Turing instability in the model without cannibalism (5), forany parameter regime.


## 2.4. Existence of Turing Instability with Cannibalism

Given that there is no Turing instability in the model without cannibalism (5), we now investigate the effect of including cannibalism, on Turing instability in the model. To this end we investigate the spatially explicit model (8)–(10). We first introduce several structure theorems from linear algebra in the literature [31–33].Definition 6 (strongly stable).
A real matrixA is said to be strongly stable if A-D is stable for all diagonal matrices D.Definition 7.
LetP denote the class of matrices whose signed principal minors are all positive and P0+ the class whose signed principal minors are all nonnegative, with one of each order positive.Theorem 8.
A3×3 real matrix A is strongly stable if and only if A∈P0+, and A is stable.Theorem 9.
A is strongly stable if and only if A is s-stable.Theorem 10.
If the kinetic system(26)dudt=fu,v,rdvdt=gu,v,rdrdt=hu,v,ris s-stable, then no Turing bifurcation is possible from the uniform steady state solution.Theorem 11.
If the kinetic system(27)dudt=fu,v,rdvdt=gu,v,rdrdt=hu,v,rcontains an unstable subsystem, then Turing bifurcation is possible from the uniform steady state solution.Remark 12.
We will now use the above theory to show that cannibalism can in fact induce Turing instability. We will first show that the Jacobian matrix of the reaction terms of the model with cannibalism (8)–(10), given by (A.2), isnot strongly stable, for certain values of the cannibalism parameter s. This will yield the result as a simple corollary to Theorems 9 and 10. Note similar ideas were used in [32], to show that a certain SEIR type epidemics model does not possess Turing instability, as long as the disease-caused death rate parameter is zero, but can have Turing patterns for positive disease-caused death rate.
Recall the Jacobian matrix of the reaction terms of the model with cannibalism is given by(28)J=r∗1-1+γsv∗11+γsu∗1u∗1r-sv∗1-m-su∗10-r∗10t1-2ur∗1-u∗1=free+++-0-0-.Remark 13.
Notice we write the sign ofJ11 as free, because due to the cannibalism parameter s the sign of J11=r∗1-1+γsv∗1 could now be positive or negative.Theorem 14.
Consider the three-species model with cannibalism (8)–(10) and its Jacobian matrix J of reaction terms, given by (28). Consider a parameter set for which J is stable, whilst the cannibalism parameter s satisfies(29)r∗1-1+γsv∗1>0,r∗1-1+γsv∗1-m-su∗1>0,orr∗1-1+γsv∗1+t1-2ur∗1-u∗1>0.
Then there exist diffusion coefficients(d1,d2,d3), such that the spatially homogenous steady state (u∗1,v∗1,r∗1) of (8)–(10) can be driven unstable due to Turing instability.The proof relies on extracting an unstable submatrix from (28). This is equivalent to some signed principal minor not belonging to P0+ and hence J given by (28) not being strongly stable.Proof.
Note if we look at the minor gotten by removing row 3 and column 3, we obtain the following submatrix:(30)r∗1-1+γsv∗11+γsu∗1r-sv∗1-m-su∗1.
This is clearly seen to be unstable ifr∗1-1+γsv∗1-m-su∗1>0. Also, if we look at the minor gotten by removing row 2 and column 2, we obtain the following submatrix:(31)r∗1-1+γsv∗1u∗1-r∗1t1-2ur∗1-u∗1.
This is clearly seen to be unstable ifr∗1-1+γsv∗1+t1-2ur∗1-u∗1>0.
Thus if either of these conditions are met,J given by (28) contains unstable submatrices. That is, certain signed principal minors are not in P0+. Thus via Theorem 8  J is not strongly stable, which implies via Theorem 9 that J is not s-stable. This then implies that there exist diffusion coefficients (d1,d2,d3) for which Turing instability is acertainty, via an application of Theorems 10 and 11. This proves the theorem.


## 3. Numerical Analysis of Turing Instability

### 3.1. The Effect of Cannibalism on Turing Instability

We now numerically demonstrate that if the conditions via Theorem14 are met, we can find diffusion coefficients d1,d2,d3 so as to induce Turing instability in (8)–(10). We choose the following parameter set, d1=10-11, d2=1.7×10-8, d3=0.9, r=0.01, m=0.1, γ=0.01, s=0.01, u=0.13, t1=0.21. For this parameter set we see that u∗1=0.058844, v∗1=0.005850, r∗1=1.162736. Thus r∗1-1+γsv∗1=0.1627>0 and r∗1-1+γsv∗1-m-su∗1=0.0614>0. Thus the conditions of Theorem 14 are met, and we can expect Turing instability. This is demonstrated in the plots next.


### 3.2. Sensitivity Analysis: Turing Space and Dispersion Plots

Since cannibalism is thesole factor in bringing about Turing instability in (8)–(10), we decide to further investigate this result. We perform next a sensitivity analysis, also referred to as a Turing space analysis, considering the effect on Turing instability with two parameters at a time. Since cannibalism is the central theme under investigation, we check the sensitivity of the cannibalism parameter againstevery other parameter. In our simulations we refer to the current literature [8, 34] to keep the parameters close to reality. In all of the plots in Figures 2 and 3 the blue area is the region where Turing instability exists, and the white area is the region where it does not. The results are presented next.We next display the dispersion relation, as we vary the cannibalism parameters.We interpret and discuss the above results in the Discussion and Conclusion (Section6).


## 3.1. The Effect of Cannibalism on Turing Instability

We now numerically demonstrate that if the conditions via Theorem14 are met, we can find diffusion coefficients d1,d2,d3 so as to induce Turing instability in (8)–(10). We choose the following parameter set, d1=10-11, d2=1.7×10-8, d3=0.9, r=0.01, m=0.1, γ=0.01, s=0.01, u=0.13, t1=0.21. For this parameter set we see that u∗1=0.058844, v∗1=0.005850, r∗1=1.162736. Thus r∗1-1+γsv∗1=0.1627>0 and r∗1-1+γsv∗1-m-su∗1=0.0614>0. Thus the conditions of Theorem 14 are met, and we can expect Turing instability. This is demonstrated in the plots next.


## 3.2. Sensitivity Analysis: Turing Space and Dispersion Plots

Since cannibalism is thesole factor in bringing about Turing instability in (8)–(10), we decide to further investigate this result. We perform next a sensitivity analysis, also referred to as a Turing space analysis, considering the effect on Turing instability with two parameters at a time. Since cannibalism is the central theme under investigation, we check the sensitivity of the cannibalism parameter againstevery other parameter. In our simulations we refer to the current literature [8, 34] to keep the parameters close to reality. In all of the plots in Figures 2 and 3 the blue area is the region where Turing instability exists, and the white area is the region where it does not. The results are presented next.We next display the dispersion relation, as we vary the cannibalism parameters.We interpret and discuss the above results in the Discussion and Conclusion (Section6).


## 4. Hopf Bifurcation

### 4.1. Hopf Bifurcation in Terms of the Cannibalism Parameter

We will now investigate the correct criterion for Hopf bifurcation in (3), in terms of the cannibalism parameter s. The criterion provided in [26] is incorrect. See Figures 6 and 7. In order to investigate the Hopf bifurcation of model system (3), we follow the method developed by Liu [35]. It can be seen that the Hopf bifurcation at s=s∗ can occur provided that A1(s∗), A3(s∗), and Φ(s∗)=A1(s∗)A2(s∗)-A3(s∗) are smooth functions of s in an open interval of s∈R such that(1)
A1(s∗)>0, A3(s∗)>0, and Φ(s∗)=A1(s∗)A2(s∗)-A3(s∗)=0;(2)
dΦ(s)/dss=s∗≠0.Thus we can state the following theorem.Theorem 15.
Under conditions (B.2), (B.3), (B.5), (B.7), and (B.9) there is a Hopf bifurcation of the positive equilibrium point (u∗1,v∗1,r∗1) at s=s∗ of model system (3), where A1(s∗)>0, A3(s∗)>0, and A1(s∗)A2(s∗)-A3(s∗)=0.The coefficientsA1, A2, and A3 and details of conditions (B.2)–(B.9) are provided in the Appendix.We now numerically illustrate a correct bifurcation point for Hopf instability. We begin by trying to find ans=s∗, such that Φ(s∗)=A1(s∗)A2(s∗)-A3(s∗)=a(s∗)2+bs∗+c=0 whilst dΦ(s)/dss=s∗=2as∗+b≠0.The form ofΦ is complicated and depends on the correctly calculated A1, A2, A3; see the Appendix. We choose the same parameters from [26], r=0.5, m=1, u=0.1, γ=1.5, t1=1.3, but the correct form of the coefficients as we have worked out in the Appendix and plot Φ(s).


## 4.1. Hopf Bifurcation in Terms of the Cannibalism Parameter

We will now investigate the correct criterion for Hopf bifurcation in (3), in terms of the cannibalism parameter s. The criterion provided in [26] is incorrect. See Figures 6 and 7. In order to investigate the Hopf bifurcation of model system (3), we follow the method developed by Liu [35]. It can be seen that the Hopf bifurcation at s=s∗ can occur provided that A1(s∗), A3(s∗), and Φ(s∗)=A1(s∗)A2(s∗)-A3(s∗) are smooth functions of s in an open interval of s∈R such that(1)
A1(s∗)>0, A3(s∗)>0, and Φ(s∗)=A1(s∗)A2(s∗)-A3(s∗)=0;(2)
dΦ(s)/dss=s∗≠0.Thus we can state the following theorem.Theorem 15.
Under conditions (B.2), (B.3), (B.5), (B.7), and (B.9) there is a Hopf bifurcation of the positive equilibrium point (u∗1,v∗1,r∗1) at s=s∗ of model system (3), where A1(s∗)>0, A3(s∗)>0, and A1(s∗)A2(s∗)-A3(s∗)=0.The coefficientsA1, A2, and A3 and details of conditions (B.2)–(B.9) are provided in the Appendix.We now numerically illustrate a correct bifurcation point for Hopf instability. We begin by trying to find ans=s∗, such that Φ(s∗)=A1(s∗)A2(s∗)-A3(s∗)=a(s∗)2+bs∗+c=0 whilst dΦ(s)/dss=s∗=2as∗+b≠0.The form ofΦ is complicated and depends on the correctly calculated A1, A2, A3; see the Appendix. We choose the same parameters from [26], r=0.5, m=1, u=0.1, γ=1.5, t1=1.3, but the correct form of the coefficients as we have worked out in the Appendix and plot Φ(s).


## 5. Cannibalism as a Life Boat Mechanism

Cannibalism often gives rise to counterintuitive phenomenon in population dynamics. Alife boat mechanism is a feature of cannibalism, where a noncannibalistic population is doomed to extinction but can persist under the act of cannibalism. We now explore such features in our model systems. We first show numerically that a life boat mechanism for the prey exists. Next we conjecture that a life boat mechanism for the predator also exists.


### 5.1. Cannibalism as a Life Boat for the Prey

Notice that, in the special case thatr=m, a prey free equilibrium (u∗1,v∗1,0) is possible in both the no cannibalism model (4) and the model with cannibalism (3). We show in Figure 8 numerically that a life boat mechanism for the prey also exists. This is not proved rigorously. A rigorous investigation will have to employ the normal form theory, for nonlinear stability analysis, as there are two zero eigenvalues at equilibrium (u∗1,v∗1,r∗1), and thus this equilibrium point is nonhyperbolic. However we can make the following conjecture.Conjecture 16.
Consider model system (3). Under certain parametric restrictions, if there is no cannibalism, that is, s=0, a small perturbation to prey free state (u∗1,v∗1,0) will still yield extinction of the prey. However, if s>0, the prey population will recover, to reach a nontrivial (u∗1,v∗1,r∗1) state.Remark 17.
What we notice is that, under certain parametric restrictions, the prey free state is globally stable, but introducing cannibalism results in coexistence of all species, and orbits converge to a nontrivial interior equilibrium(u∗1,v∗1,r∗1). This shows that a life boat mechanism exists for the prey as well. Also see Figure 8.


### 5.2. Cannibalism as a Life Boat for the Predator

Conjecture 18.
Consider model system (3). Under certain parametric restrictions there exist cannibalism rates γ,s such that if there is no cannibalism, that is, s=0, a small perturbation to the predators in the (0,0,t1/u) state will still yield extinction of the predator. However, if s>0, the predator population will recover, to reach a nontrivial (u∗1,v∗1,r∗1) state.Remark 19.
The Jacobian matrix for the no cannibalism model (4) at 0,0,t1/u [26] is given by(32)J=t1u-110r-m0-t1u0-t1.Satisfaction of the following conditions,(33)u>t1,r<m1-t1u,are sufficient to satisfy the Ruth-Hurwitz criteria for local asymptotic stability of 0,0,t1/u. Thus if there exist cannibalism rates γ,s such that (u∗1,v∗1,r∗1), the nontrivial equilibrium of the model with cannibalism (3), is globally stable and (33) also holds, then the populations will rebound from (ϵ,ϵ,t1/u+ϵ) and be driven up to (u∗1,v∗1,r∗1). This tells us that a small amount of adult and predator populations without cannibalismcannot sustain themselves under certain parametric restrictions, as (0,0,t/u) is locally stable. Thus (ϵ,ϵ,t/u+ϵ) will be pulled back to (0,0,t/u). However, the act of cannibalism, under these parametric restrictions (assuming one can formally prove global stability of the equilibrium), can cause these small amounts of adult and juvenile predator populations to rise, and so (ϵ,ϵ,t/u+ϵ) can be driven up to (u∗1,v∗1,r∗1).


## 5.1. Cannibalism as a Life Boat for the Prey

Notice that, in the special case thatr=m, a prey free equilibrium (u∗1,v∗1,0) is possible in both the no cannibalism model (4) and the model with cannibalism (3). We show in Figure 8 numerically that a life boat mechanism for the prey also exists. This is not proved rigorously. A rigorous investigation will have to employ the normal form theory, for nonlinear stability analysis, as there are two zero eigenvalues at equilibrium (u∗1,v∗1,r∗1), and thus this equilibrium point is nonhyperbolic. However we can make the following conjecture.Conjecture 16.
Consider model system (3). Under certain parametric restrictions, if there is no cannibalism, that is, s=0, a small perturbation to prey free state (u∗1,v∗1,0) will still yield extinction of the prey. However, if s>0, the prey population will recover, to reach a nontrivial (u∗1,v∗1,r∗1) state.Remark 17.
What we notice is that, under certain parametric restrictions, the prey free state is globally stable, but introducing cannibalism results in coexistence of all species, and orbits converge to a nontrivial interior equilibrium(u∗1,v∗1,r∗1). This shows that a life boat mechanism exists for the prey as well. Also see Figure 8.


## 5.2. Cannibalism as a Life Boat for the Predator

Conjecture 18.
Consider model system (3). Under certain parametric restrictions there exist cannibalism rates γ,s such that if there is no cannibalism, that is, s=0, a small perturbation to the predators in the (0,0,t1/u) state will still yield extinction of the predator. However, if s>0, the predator population will recover, to reach a nontrivial (u∗1,v∗1,r∗1) state.Remark 19.
The Jacobian matrix for the no cannibalism model (4) at 0,0,t1/u [26] is given by(32)J=t1u-110r-m0-t1u0-t1.Satisfaction of the following conditions,(33)u>t1,r<m1-t1u,are sufficient to satisfy the Ruth-Hurwitz criteria for local asymptotic stability of 0,0,t1/u. Thus if there exist cannibalism rates γ,s such that (u∗1,v∗1,r∗1), the nontrivial equilibrium of the model with cannibalism (3), is globally stable and (33) also holds, then the populations will rebound from (ϵ,ϵ,t1/u+ϵ) and be driven up to (u∗1,v∗1,r∗1). This tells us that a small amount of adult and predator populations without cannibalismcannot sustain themselves under certain parametric restrictions, as (0,0,t/u) is locally stable. Thus (ϵ,ϵ,t/u+ϵ) will be pulled back to (0,0,t/u). However, the act of cannibalism, under these parametric restrictions (assuming one can formally prove global stability of the equilibrium), can cause these small amounts of adult and juvenile predator populations to rise, and so (ϵ,ϵ,t/u+ϵ) can be driven up to (u∗1,v∗1,r∗1).


## 6. Discussion and Conclusion

In the current manuscript, to the best of our knowledge, we report thefirst cannibalism induced Turing instability result, in spatially explicit three-species age structured predator-prey systems. That is, if one considers the spatially explicit version of the three-species age structured ODE model originally proposed in [18], that is, (5), then without cannibalism there can be no Turing instability inany parameter regime. This is shown via Theorem 4. However,introducing cannibalism brings about Turing instability, in (8)–(10), seen via Theorem 14. Also note that although the Hopf bifurcation analysis on the model originally proposed by Magnússon has been done in [26], the results therein are incorrect. We derive the correct conditions for the Hopf bifurcation via Theorem 15. In the case of the prey species, we see numerical evidence of a life boat via Figure 1.Figure 1
Here we demonstrate that cannibalism can indeed induce Turing instability in (8)–(10). We see a spatial Turing pattern in the juvenile species in the left panel. The right panel shows the spatial profile of all three species, after the simulation is run much longer (t=1000). We see Turing persists in both the adult and juvenile predators, but the prey has very weak Turing. The values of the parameters here are d1=10-11, d2=1.7×10-8, d3=0.9, r=0.01, m=0.1, γ=0.01, s=0.01, u=0.13, t1=0.21. For this parameter set we see that u∗1=0.058844, v∗1=0.005850, r∗1=1.162736.Figure 2
The values of the parameters here ared1=10-11, d2=1.7×10-8, d3=0.9, γ=0.01, m=0.1, u=0.13, t1=0.21. We look at the Turing space (sensitivity) generated by varying s and r, each in the range [0,1].Figure 3
The values of the parameters in (a) ared1=10-11, d2=1.7×10-8, d3=0.9, γ=0.01, m=0.1, r=0.01, u=0.13. We look at the Turing space (sensitivity) generated by varying s and u, each in the range [0,1]. The values of the parameters in (b) are d1=10-11, d2=1.7×10-8, d3=0.9, γ=0.01, r=0.1, u=0.13, m=0.1. We look at the Turing space (sensitivity) generated by varying s and t1, each in the range [0,1].
(a) (b)From an ecological point of view, Theorem14 tells us that, for thisspecific model, cannibalism induces Turing instability. Thus in order to investigate the Turing instability result further, we perform a sensitivity analysis, where we look at the Turing space generated by the cannibalism parameter, withevery other parameter in the system.We look at how the Turing space changes as we varys the cannibalism parameter and γ the rate of intake by the cannibalistic adult. We see that Turing occurseverywhere in s∈[0,1],γ∈[0,1], so we do not show a plot of this case. This tells us that the interplay between the amount of cannibalism and intake rate of the cannibalalways produces Turing instability, in this range (to keep things realistic we vary the parameters only in [0,1]). That is, if we keep s fixed and increase or decrease γ, the Turing instability persists.In Figure2 we look at how the Turing space changes with respect to the interactions between the cannibalistic adult predator and the juvenile predator. Here we vary s the cannibalism parameter and r the growth rate of the cannibalistic adult that adds to the juvenile population. We see that Turing occurs for all s∈[0,1], if r is very low, say r∈(0,0.03]. For low values of s, say s∈(0,0.2], Turing occurs if we increase r up to about 0.2, but not if we further increase r. Furthermore, we observe that we lose Turing, for moderate values of r, say about 0.2 or more, if we increase s past 0.5. This tells us that the interplay between the amount of cannibalism and growth rate of the cannibalistic adult essentially always acts to remove Turing for values of s>0.5. However Turing essentially persists for all s, if say r∈(0,0.03] that is very low. Now we vary s the cannibalism parameter and m the loss in the juvenile population, due to natural death as well as transitioning into the adult class. We see that Turing occurs almost everywhere except for very low values of s,m, that is, s∈(0,0.3] and m∈(0,0.1]. This tells us that the interplay between the amount of cannibalism and loss rate of the juvenile essentially always acts to give Turing for most values in the considered range, so again we do not plot this case.In Figure3 we look at how the Turing space changes with respect to the interactions between the cannibalistic adult predator and the prey species. In Figure 3(a) we vary s the cannibalism parameter and u the growth rate of the prey. This particular interaction is probably the most interesting in our opinion. As a first analysis we noticed that Turing occurs for all s∈[0,1], for a narrow band of u values, say u∈[0.15,0.22]. However, upon further discretization of the parameter space, we observed that there are disconnected patches where Turing occurs, for roughly s∈[0.65,0.75] and u∈[0.55,0.6], and then again for certain values of u∈[0.7,1]. This property is not seen in any of the other simulations, despite various further discretization in all of the parameters. In Figure 3(b) we vary s the cannibalism parameter and t1 the competition coefficient for the prey. Note the competition mediated effects of cannibalism have been the focus of a number of recent studies [8, 36]. We see that Turing occurs for all s∈[0,1], for a narrow band of t1 values, say t1∈[0.11,0.2]. Also Turing occurs for essentially all t1 values, say t1≥0.1, for a narrow band of s values, s∈[0.95,1]. This tells us that the interplay between the amount of cannibalism and competition amongst prey essentially acts to give Turing for restricted values of competition parameter t1.In the same vein as the above, we also conduct a dispersion relation analysis. Here we plot the coefficientA1A2-A3 for various values of the wave number k. Recall that Turing instability occurs, if this coefficient dips below the k axis. We construct a sequence of 4 graphs, by varying the cannibalism parameter s. What we notice is that as s is increased (for other parametric values fixed as in Figure 1), we see the coefficient moves upwards and then becomes positive past s=0.31. This tells us that the effect of increasing cannibalism here is to remove the Turing instability.In light of our results we can summarise a few important findings.(1)
In the spatially explicit form of the classical model considered by Magnússon and then rescaled by Tang, cannibalismcan cause Turing instability.(2)
The Turing instability induced by cannibalism seems to be most sensitive to the prey growth rate parameteru and t1 the competition coefficient for the prey. It also seems to be least sensitive to γ the rate of intake by the cannibalistic adult and m the loss rate in the juvenile population. This tells us that the adult predator-prey relation is critical, despite the adult being able to cannibalise the juvenile predator.(3)
Magnússon and Tang’s findings are that, in the ODE model, increasing the cannibalism parameters destabilises the system. Given that we choose similar parameters, our result suggests that, in the spatially explicit case, increasing the cannibalism parameter s removes the Turing instability and hencestabilises the system; see Figure 4. This makes the ODE and PDE models quitedifferent dynamically.Figure 4
Here we plot the dispersion relation of the coefficientA1A2-A3 versus the wave numbers k, for various values of the parameter s. Recall there is Turing instability when at least some part of the coefficient A1A2-A3 lies below the k axis.Figure 5
Here we see a plot ofΦ(s) versus s for the other parameters as mentioned earlier from [26], r=0.5, m=1, u=0.1, γ=1.5, t1=1.3. We clearly see that dΦ(s)/dss=1.001>0, and dΦ(s)/dss=0.75=0. The root occurs at approximately s=1.001.Figure 6
Here we take the parameters as mentioned earlier from [26], r=0.5, m=1, u=0.1, γ=1.5, t1=1.3, s=0.9. We clearly see that, at s=0.9, Φ<0, from Figure 5, and we expect limit cycle dynamics, which we see in (a). We compare to the simulation from [26] in (b), which shows a stable spiral at s=0.9, which is clearly incorrect.
(a) (b)Figure 7
Here we take the parameters as mentioned earlier from [26], r=0.5, m=1, u=0.1, γ=1.5, t1=1.3. With s=0.986494, Φ<0, from Figure 5, and we clearly see limit cycle dynamics, as expected in (a). Now if we take s=1.0141, where Φ>0, from Figure 5, we see a stable spiral, again as expected, in (b).
(a) (b)Figure 8
Here we demonstrate that cannibalism can act as a lifeboat for the prey species also. The values of the parameters here arer=0.01, m=0.01. γ=0.01, u=0.062, t1=0.09. We see in (a) that, with s=0, or no cannibalism, the prey is driven to extinction. However, in (b), we see that introducing a little bit of cannibalism, that is, by choosing s=0.1, the prey population immediately rebounds to a high level.
(a) (b)Note the Turing patterns we currently obtain are purely spatial. It would be interesting to investigate if spatiotemporal patterns driven by Turing-Hopf type instabilities exist. If one can prove otherwise, it raises further interesting questions into the nature of cannibalism and its ecological consequences, such as why would conspecific consumption hinder temporal changes in the population, at fixed locations in space? The other interesting question for possible future investigation is considering cannibalism among juveniles. We know for a fact that this does take place in certain amphibian species [36]. One might also consider the effect of cannibalism occurring simultaneously amongst adults and juveniles. Lastly, akin to [17] one can investigate the occurrence of cannibalism in the prey. Further investigation of the disconnected Turing space seen in Figure 3 should also be a future direction. The issue of whether cannibalism acts as a life boat mechanism for the predator, in (3), remains unproven and can also be considered in future. All in all, we hope that the current work leads to further investigations and collaborations into the fascinating effects of cannibalism and the oftencounterintuitive role it plays in natural predator-prey communities.



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