STABILITY AND OPTIMAL CONTROL FOR SOME CLASSES OF TRITROPHIC SYSTEMS

The objective of this paper is to study an optimal resource management problem for some classes of tritrophic systems composed by autotrophic resources (plants), bottom level consumers (herbivores) and top level consumers (humans). The first class of systems we discuss are linear chains, in which biomass flows from plants to herbivores, and from herbivores to humans. In the second class of systems humans are omnivorous and hence compete with herbivores for plant resources. Finally, in the third class of systems humans are omnivorous, but the plant resources are partitioned so that humans and herbivores do not complete for the same ones. The three trophic chains are expressed as Lotka-Volterra models, which seems to be a suitable choice in contexts where there is a shortage of food for the consumers. Our model parameters are taken from the literature on agro-pastoral systems in Sub-Saharan Africa.


1.
Introduction.The scientific literature on natural resource exploitation is displaying a steady growth and a number of new research directions have been explored.One of the central topics to attract the interest of the community is represented today by the analysis of the qualities and effects of human interactions with the environment, which requires to be adequately modeled in many different practical contexts [23] considering its fundamental implications on ecosystem balancing, social welfare and economic growth [3].Trophic systems that include humans were classified in [11] as a relevant category of ecosystems from the management viewpoint and experiments have been conducted to characterize the ecological role of humans in specific biological communities [4].The impact of human development on sustainability is also a central topic in ecological economics studies [28].

LUCA GALBUSERA, SARA PASQUALI AND GIANNI GILIOLI
In recent times, control theory has been used to support resource management decisions.Relevant examples include harvesting [21,39], fishery management [5,30,8], pest control [6,31].In many of the references reported above, the reader can observe a trend towards the synthesis of optimal resource exploitation strategies.In particular, a number of papers have considered how optimal control theory can be used to manage trophic chains.For instance, in [22] the optimal harvesting strategy of a single growing species was addressed.In [32], an infinite-horizon optimal control problem with discount was proposed, consisting in the maximization of a combination between the human population level and the associated welfare status over time.In [10], optimal control theory was applied to stabilization and synchronization problems in Lotka-Volterra models.Optimal foraging was discussed in [19] within the context of two-prey-one-predator populations showing adaptive behaviors.Optimization methods for solving optimal control problems involving Lotka-Volterra models were investigated in [35] under special restrictions on the time dependent control functions.In [1] the problem of maximizing the total population in a Lotka-Volterra tritrophic system was studied and the optimal species separation was characterized in terms of the model parameters.In [36], optimal control was applied to a Rosenzweig-MacArthur tritrophic system in order to achieve sustainable management strategies.These results were extended to stochastic systems in [37].Reference [2] deals with the optimal management of two species accounting for various types of functional responses describing predator-prey interactions.
In this paper we consider how optimal resource management policies depend on special structures of the trophic chain and investigate infinite time horizon optimal control problems with the objective of characterizing management strategies that sustain the human population's biomass and promote welfare.In particular, • we consider a fully tritrophic system composed by plants, herbivores and humans.The modeling of the plant resources accounts for the importance of the primary production for the sustainability of agro-pastoral systems in fragile ecosystems like most of the arid and semi-arid areas where pastoralism in Africa is present; • moreover, the diversity of the trophic structure characterizing the agro-pastoral transition is analyzed.According to [7], in the household economy of the traditional pastoral system humans act as secondary consumers and only a limited amount of energy comes from the plants.Under the effects of many ecological, economic and social drivers [9,33] traditional agro-pastoral communities have changed their habits towards the sedentarization and the integration of smallholder farm systems into the grassland.The trophic chain sustaining these communities becomes more complex and diversified.Different management strategies of the grassland and cropping systems (e.g., pure pasture or mixed stands, choice of the level of exploitation) lead to different trophic structures of the ecological system supporting the agro-pastoral community and are expected to put in place different challenges to the sustainability of the agro-pastoral system.The analysis we perform deals with stability properties of these trophic chains and aims at the definition of sustainable level of exploitation of resources.
A representation of the system under analysis, which takes into account the trophic interactions between the resources and the human population together with its welfare status, is reported in Figure 1.Therein ξ 1 , ξ 2 , y, z ≥ 0 are the biomasses of two plant sources, the herbivores and the human population, respectively, and w ≥ 0 is a measure of the level of human welfare.Parameters q y , q z > 0 are the specific loss rates of organic matter due to excretion, death and respiration at the upper levels of the trophic chain, represented by herbivores and humans.Quantity C ≥ 0 is a measure of the specific human biomass consumption rate for non-vital activities not directly related to trophic processes (e.g., adaptation and activities related to welfare).This graphical representation puts in evidence that a fraction of the human's harvest is devoted to increasing the biomass of the human population, while the rest is spent to improve the quality of life.Moreover, it is also assumed that a portion Cz of the human biomass is lost for non-vital activities, as well.
The main objective of this paper is to analyze this system taking into account different trophic relationships between humans and their resources, as it will be illustrated in the following.Thus, for plainness we will not include the level of welfare as a state variable characterizing the system, while a human biomass consumption that can be interpreted a support to non-vital activities is maintained in the balance equations, as proposed in [32,14].Following these references, the human benefit achieved through the consumption term Cz is expressed by a utility function W (C), which is assumed to fulfill the following assumptions [32] with ( ) and ( ) denoting the first and second derivatives with respect to the argument.According to the latter simplifying assumption, a general structure for the different classes of tritrophic systems we will consider is provided by the following set of ODEs, representing two interacting subsystems S1 and S2 : together with the initial condition 2 ) Parameter r i , for i = 1, 2, represents the specific growth rate of the i-th source and K i is the carrying capacity.Trophic interactions are modeled by means of the functional response f i1i2 (s), for (i 1 , i 2 ) ∈ {(y, ξ 1 ), (z, ξ 1 ), (z, y), (z, ξ 2 )}, which is assumed to be concave and bounded ([41], p. 87).In the applications, Holling-type II or Ivlev models are commonly used [16,15].The corresponding terms a i1i2 > 0 are related to the efficiency of the predation process.Parameters θ yξ1 , θ zξ1 , θ zy , θ zξ2 ∈ (0, 1] are the biomass conversion factors.Finally, the biomass flow across levels also depends on quantities Dyξ1 , Dzξ1 , Dzy and Dzξ2 , which describe the food demands of consumers per time unit.In general, we assume Dyξ1 > 0 and Dzξ1 , Dzy , Dzξ2 ≥ 0. From the structure of system S1 in (2) it can be observed that various types of trophic chains can be accounted for, including cases in which humans act as omnivores.Furthermore, the presence of subsystem S2 enables us to consider the effects of the availability of a complementary food source for the top level consumer.This aspect seems to be relevant both in a modeling perspective and for control purposes, as emphasized in the recent literature [38,18,25,40,26,17].
The rest of this paper is organized as follows: in Section 2 we specify the Lotka-Volterra approximation of system (2), show positiveness and boundedness of the state trajectories, and identify a positively invariant and attractive set.In Section 3 we characterize the non-coexistence equilibrium states and in Section 4 the coexistence equilibrium states.In Section 5 we examine the stability of the coexistence equilibrium states.In Section 6 we present an optimal control problem and characterize some relevant associated properties.Section 7 contains some numerical examples.
2. Trophic chain models.We study a class of models obtained from (2) through a linear approximation of functions f i1i2 (•): where System ( 3) is a Lotka-Volterra model; the use of the approximation (4) is feasible when the biomass of the preyed species is very limited, as it is the case in the application we are addressing.In this paper, all parameters of model (3) will be assumed to be assigned constants with the notable exception of C, D zξ1 , D zy and D zξ2 , which represent the set of control variables whose value along time has to be chosen in order to maximize an objective functional.As the discussion of our results on optimal control is postponed to Section 6, until then these quantities are assumed to be constant over time as well, in order to perform an analysis of the equilibrium states and the associated stability properties.
For future utility, introduce the following notation, with i = 1, 2: It is assumed that these quantities are well defined each time they are used.
System (3) is a general representation capturing, as special cases, the three trophic chains we are interested in (see Figure 2), namely 1. linear chain (S l ): we consider only subsystem S 1 and assume D zξ1 , D zξ2 = 0 and D zy > 0; 2. food chain with omnivory (S o ): we consider only subsystem S 1 and assume D zξ2 = 0, and D zξ1 , D zy > 0; 3. food chain with omnivory and source partition (S p ): we consider the composition of subsystems S 1 and S 2 and assume D zξ1 = 0 and D zy , D zξ2 > 0. In our perspective, the different structures described above can serve as a simplified representation of different practical situations.In particular, the linear chain represents the humans as consumers of food drawn from an animal source; as such, it will be referred to as an approximate description of traditional pastoral systems.The other cases model humans as omnivores, and hence we refer to them as representations of an agro-pastoral system.In particular, in the case of S p a source partition is introduced in order to better model the situation in which herbivores and humans access separate vegetable food sources, i.e. forage and crops, respectively.
The notations introduced above and Theorem 2.1 are useful in defining parameter regimes with non-trivial dynamics.First recall that, from point 2 in Theorem 2.1, it results lim sup t→∞ ξ i (t) ≤ K i , i = 1, 2. Consequently, since we are assuming that all the model parameters (including the controls) are constant over time, to avoid trivial dynamics starting from (ξ 0 1 , y 0 , z 0 , ξ 0 2 ) > 0 we have to rule out the condition θ yξ1 D yξ1 K 1 − q y < 0, which would imply lim t→∞ y(t) = 0 in view of the second equation in (3).This would also imply then lim t→∞ z(t) = 0 also in these cases.In conclusion, from now on we make the following assumptions for S l e yξ1 < 1 and e zξ1 < 1, for S o e yξ1 < 1 and e zξ2 < 1, for S p (5) which are necessary in order to avoid trivial dynamics, leading to lim t→∞ y(t) = 0 and/or lim t→∞ z(t) = 0.
The next three sections are devoted to the existence and stability analysis of the equilibrium states of system (3) in its special forms.The stability properties are determined by a local analysis based on the system's Jacobian, reported in Appendix B, as well as on Lyapunov functions, in some cases.

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3. Non-coexistence equilibrium states and their stability.Non-coexistence equilibrium states of system (3) are defined as states where at least one species goes extinct.We now provide a list of the non-coexistence equilibrium states of systems S l , S o and S p : The associated local stability properties are derived in Appendix C and summarized in Table 1.Observe that the positiveness of many terms appearing in the equilibrium solutions enumerated above is implied by conditions (5) for non-trivial dynamics.
4. Coexistence equilibrium states.In a coexistence equilibrium state the abundance of all species is greater than zero.In this section we enumerate the coexistence equilibrium states of systems S l , S o and S p and give conditions for their uniqueness and positiveness.
The coexistence equilibrium states associated to model ( 3) are solutions to the algebraic system of equations LAS on (0, y, z) LAS on (0, y, z, 0)

LAS when
LAS when unstable Table 1.Stability properties of the non-coexistence equilibrium states (LAS=locally asymptotically stable). where Term ∆ 1 is zero for both S l and S o , while term ∆ 2 is zero for both S l and S p .It follows that the coexistence equilibrium is unconditionally unique for S l and S p , while for S o uniqueness requires the assumption ∆ 2 + ∆ 3 = 0.The coexistence equilibrium values of the state variables will be denoted by ξ * 1 , y * , z * and ξ * 2 , assuming that they are uniquely defined.
1. System S l The coexistence equilibrium for system S l is 1 − e zy g yξ1 − 1 The equilibrium E l ξ1yz is positive when 1 − ezy g yξ 1 − e yξ1 > 0, i.e.
Observe that E l ξ1y is unstable when this condition holds, see Table 1.

System S o
The following coexistence equilibrium can be found in this case: where The considered equilibrium is unique if For the positiveness of y * and z * respectively, we need to impose the following conditions: Observe that (7) also implies the positiveness of ξ * 1 .Furthermore, the two conditions ( 6) and ( 7) imply the instability of E o ξ1z and E o ξ1y , respectively.Vice versa, when A o < 0 we obtain the reverse, which means that the coexistence equilibrium is positive iff E o ξ1z and E o ξ1y are LAS.

System S p
The following unique coexistence equilibrium can be found for S p : where ξ * 1 and ξ * 2 are the solution of the following algebraic system: where A p = qy g zξ 2 DzyK1e yξ 1 Observe that such a system admits a unique solution under our assumptions.Furthermore, the positiveness of ξ * 1 is unconditional, while ξ * 2 is positive iff K 2 g zξ2 D zy e yξ1 g yξ1 + q y (1 + g yξ1 (e yξ1 − 1)) > 0 Finally, for the positiveness of y * and z * the following conditions need to be fulfilled, respectively: Stability properties of the coexistence equilibrium states.In this subsection we give a characterization of the (global) stability properties of the coexistence equilibrium states for systems S l , S o and S p .In defining a solution globally asymptotically stable (GAS) we only consider the asymptotic behavior of trajectories originating in the strictly positive orthant.As a preliminary remark we mention that, differently from the cases of S l and S p , the stability analysis of S o is complicated by the presence of two paths connecting the same source to the top consumer and partitioning the corresponding biomass flow.
In order to study the stability properties of system (3), we introduce the following Lyapunov function: where c ξ1 , c y , c z and c ξ2 are arbitrary positive constants.The derivative of the considered Lyapunov function along the trajectories of system ( 3) is given by . Now, for symmetric matrices, denote by symbol ( ) each of its symmetric blocks.Then, the latter equation can be rewritten as where We are now in a position to analyze the global asymptotic stability of the coexistence equilibrium states associated to each type of system.

System S l
We have to evaluate the corresponding components of V in (8) in the state STABILITY & OPTIMAL CONTROL FOR SOME CLASSES OF TRITROPHIC SYST.267 space (ξ 1 , y, z).Matrix M 1 reduces to the following form M l 1 : It is easy to show that c ξ1 , c y , c z > 0 can be chosen so that m yξ1 = m zy = 0. Thus, the resulting derivative of the Lyapunov function along the trajectories of S l is equal to The following implications hold: so that the associated invariant set is (ξ * 1 , y * , z * ).Thus, E l ξ1yz is GAS.

System S o
Matrix M 1 reduces to the following form: In order to prove that V < 0 we impose that the cross-product terms are zero, i.e. m yξ1 = m zξ1 = m zy = 0.It is always possible to define c ξ1 , c y , c z > 0 solving the problem when the following condition holds: This assumption ensures that the cross-product terms appearing in the Lyapunov function's derivative are zero.The efficiency gap θ zξ1 − θ zy θ yξ1 plays a fundamental role in determining the stability properties S o .To better emphasize this point, it is possible to perform a local stability analysis of the coexistence equilibrium of S o , which results in the following characteristic polynomial: Using the Routh-Hurwitz theorem, we conclude that the following condition implies that the coexistence equilibrium is LAS: We refer to the following matrix M p , obtained from M as a particular case: It is easy to show that c ξ1 , c y , c z , c ξ2 > 0 can be chosen so that m yξ1 = m zy = m zξ2 = 0.The resulting derivative of the Lyapunov function along the trajectories of S p is The following implications hold: Thus the associated invariant set reduces to (ξ * 1 , y * , z * , ξ * 2 ) and we conclude that E p ξ1yzξ2 is GAS.
6. Optimal control.In this section we introduce an optimal control problem associated to system (3).The objective of our control strategies will be to maximize a suitable societal objective functional in order to enhance the human population level z and the utility function W (C), defined according to the assumptions (1).In our setup, from now on quantities C, D zξ1 , D zξ2 and D zy are considered control variables whose value along time has to be assigned so to maximize the following societal objective function [32,14]: subject to the system of ODEs (3) and In this formulation, e −δt is the discount term, with δ > 0 fixed [42], and the positive constants D ub zξ1 , D ub zy and D ub zξ2 express technical/technological limitations encountered in the human food catching process and are assumed to be constant in time.Observe that the effective number of control variables involved in the management of the trophic system, as well as the dimension of the ODE system itself, depends on the specific food chain under consideration.In particular, the control variables C and D zy are common to all cases, while each of the chains S o and S p includes an additional control variable, either D zξ1 or D zξ2 .Therefore, all the trophic chains we are considering have a so-called top-down structure from the control point of view, i.e. the control variables are associated to the regulation of the behavior of the top level consumer.
In order to tackle the optimal control problem (9), we apply the Pontryagin maximum principle.To do so, we reformulate problem (9) as the maximization of the Hamiltonian H = e −δt zW (C) + λ T F (10) where λ = [λ 1 λ 2 λ 3 λ 4 ] T is the costate vector and F = [F ξ1 F y F z F ξ2 ] T collects the right-hand sides of the equations in system (3).Now let X = [ξ 1 , y, z, ξ 2 ] T , U = [C, D zξ1 , D zξ2 , D zy ] T and define the following Hamiltonian system associated to the optimal control problem under analysis:

STABILITY & OPTIMAL CONTROL FOR SOME CLASSES OF TRITROPHIC SYST. 269
where Clearly, this system is defined in structure and size by the specific type of trophic chain we are considering.We report next the necessary conditions for an optimal solution according to the Pontryagin maximum principle, assuming that such an optimum exists: where u i is the i-th component of vector U .Observe that, when the dynamical solution of the optimal control problem ( 9) is considered, singularity is found with respect to the control variables D zξ1 , D zy and D zξ2 .This can be easily observed by writing the current-value Hamiltonian H c associated to (10) as follows: where It results that, if an optimal solution is such that some components of α(λ, ξ 1 , y, z, ξ 2 ) are zero for t ∈ [t 1 , t 2 ], t 1 < t 2 , the state trajectory has a singular arc.When this does not happen, the time evolution of the optimal control variables D zξ1 , D zy and D zξ2 is bang-bang, i.e.D zξ1 ∈ {0, D ub zξ1 }, D zy ∈ {0, D ub zy }, and D zξ2 ∈ {0, D ub zξ2 }.From now on in this paper, we focus on giving a characterization of the optimal static control policies for S l , S o and S p , consisting in the constant controls associated to the optimal final steady-states for the dynamic optimal control problem (9).To this end, we also neglect the upper bounds D ub zξ1 , D ub zξ2 and D ub zy on the control variables.In studying the optimal static control policies, we refer to the asymptotic behavior of the optimal state trajectories encountered in (9).As emphasized in [12], this kind of analysis can be very informative and enable us to evaluate the sensitivity of the optimal limit points of the state trajectories to relevant parameters of the optimization problem, such as the discount rate δ.

System S l
The adjoint system reduces to where λ l = [λ l 1 , λ l 2 , λ l 3 ] T ; recall that, in this case, D zξ1 = 0 by assumption.In order to compute the optimal static solution, we first manipulate system (13), obtaining the following ODE in λ l 2 by elimination: with The associated complete solution is of the form 21 , M l 22 and M l 23 are arbitrary constants, and coefficients µ 2i , i = 1, 2, 3 are solutions of the auxiliary equation for ( 14), namely µ 3 + α l 21 µ 2 + α l 22 µ + α l 23 = 0 By studying the latter equation it is possible to conclude that, in order to have lim t→∞ λ l 2 = 0, as requested in (12) for optimality, we must impose M l 21 = M l 22 = M l 23 = 0. We can proceed similarly for λ l 3 , obtaining the following ODE by elimination: with The complete solution, in this case, has the form λ l 3 = M l 31 e µ31t + M l 32 e µ32t + M l 33 e µ33t + N l 3 e −δt W (C) where M l 31 , M l 32 and M l 33 are arbitrary constants, and coefficients µ 3i , i = 1, 2, 3 solve the auxiliary equation for (15), which is Similarly to the previous case, it is possible to verify that the optimality condition lim t→∞ λ l 3 = 0 is fulfilled if and only if M l 31 = M l 32 = M l 33 = 0. Thanks to the calculations reported above, we can now deal with the optimality condition (11).Since we are considering optimal coexistence equilibrium states and thus y, z > 0 by assumption, it results Substituting λ l 2 and λ l 3 therein according to the expressions found above, we obtain the following algebraic optimality conditions in the variables C, D zy , ξ 1 , y and z: The adjoint system in this case is where T and, differently from the previous case, D zξ1 > 0 by assumption.
Operating on (17) by elimination, we obtain the following ODE in λ o j , j = 1, 2, 3: with The associated complete solution is of the form where M o ji , i = 1, . . ., 3, j = 1, 2, 3 are arbitrary constants, while and coefficients µ ji , i = 1, . . ., 4, j = 1, 3 are solutions of the auxiliary equation for (18), namely By some computation it results that, in order to have lim t→∞ λ o j = 0, as requested in (12) for optimality, we impose M o ji = 0, i = 1, . . ., 4, j = 1, 2, 3. Thanks to the calculations reported above, we can now deal with the optimality condition (11).Since we are considering optimal coexistence equilibrium states and thus y, z > 0 by assumption, it results 2 and λ o 3 therein according to the expressions found above, we obtain the following algebraic optimality conditions in the variables C, D zξ1 , D zy , ξ 1 , y and z: The adjoint system in this case is where T and, in A H , D zξ1 = 0 by assumption.Similarly to the previous cases, in order to compute the optimal static solution we manipulate system (19) by elimination.In this way, we obtain the following ODE in λ p j , with j = 2, 3, 4: with The associated complete solution is of the form where M p ji , i = 1, . . ., 4, j = 2, 3, 4 are arbitrary constants, while and coefficients µ ji , i = 1, . . ., 4 are solutions of the auxiliary equation for (20), namely By some computation it results that, in order to have lim t→∞ λ p j = 0, as requested in (12) for optimality, we must have M p 2i = 0, i = 1, . . ., 4, j = 2, 3, 4. It is now possible to address the optimality condition (11).Since we are considering optimal coexistence equilibrium states and thus y, z > 0 by assumption, it results Substituting λ p 2 , λ p 3 and λ p 4 therein according to the expressions found above, we obtain the following algebraic optimality conditions in the variables C, D zy , D zξ2 , ξ 1 , y, z and ξ 2 : Numerical examples.This section presents some numerical studies related to the optimal control problem discussed in the previous section.Specifically, we assume W (C) = C a C +C , where a C > 0 is a constant, and focus on the optimal static solutions of systems S l and S p as functions of δ.Our simulations are based on a parameter set derived from the literature and reported in the following Observe that this parameter set fulfills the condition related to e yξ1 for non-triviality in (5) (observe that this is the only non-triviality condition purely dependent on data and not on controls), since it results e yξ1 ≈ 0.35 < 1.
We begin by considering S l .By solving the algebraic system ( 16) parametrically with respect to δ, we obtain the results reported in Figure 3.Some interesting discussion can be made based on these solutions.To this end, recall that present utility prevails over long-run profits in determining the value of the societal objective function, when the value of δ increases.Inversely, for low values of the parameter the human population level and consumption are limited in order to preserve trophic resources useful for human consumption.This is confirmed by our numerical analysis: consumption C increases with δ and is supported by the simultaneous increase of D zy .The high level of exploitation of the herbivore biomass by humans for high values of δ reduces the relative static optimal equilibrium value.
As a second step, we study the case of S p and solve the algebraic system (21) as a function of parameter δ.The outcomes of this procedure are displayed in Figure 4.In particular, observe the ability of this system's configuration to allow Figure 3. System S l : optimal steady states and control inputs as functions of δ.
higher values of z with respect to the case of S l discussed above, thanks to the support now provided by the second plant source.
8. Concluding remarks.In this paper we examined three Lotka-Volterra tritrophic chains: a linear chain S l , a trophic chain with omnivory S o and a trophic chain with omnivory and source partition S p .We identified the equilibrium solutions of each system and also analyzed the related stability properties.In the case of systems S l and S p , we showed GAS using a standard Lyapunov method.In studying the stability of S o , we discussed the special role played by θ zξ1 − θ zy θ yξ1 , which represents the efficiency gap between the two trophic channels humans exploit to draw biomass from the food source.We also formulated an optimal control problem to determine the rates at which humans consume biomass from the lower levels of the trophic chain and the rate at which humans lose biomass to gain utility, in order to enhance a combination of utility and human biomass over an infinite time horizon.
Using data from Sub-Saharan Africa, we gave some numerical characterizations about how stationary optimal solutions vary with respect to parameter δ.It is interesting to observe how these relationships depend qualitatively on the model's structure.Specifically, comparing the case of S p with S l , we appreciated the effect of a second plant food source on the ability to maintain higher human biomass levels with respect to parameter δ.
This property is a direct consequence of a standard comparison theorem and is easily demonstrated referring to the differential inequality ξi ≤ r i ξ i (1 − ξi Ki ) characterizing the time evolution of the state variables ξ i , for i = 1, 2 and applying the Gronwall's inequality.
Step 2: 0 ≤ ξ 1 (t) + y(t) The following inequalities hold: As a consequence, by Gronwall's inequality, ∀t ≥ 0 we have Step 3: The following inequality holds in this case: and the conclusion follows by the same procedure as above.
Step 1: lim sup t→+∞ ξ i (t) ≤ K i , i = 1, 2. Also in this case, this property is a direct consequence of a standard method analogue to the one used in Step 1 of Part 1, having observed that the solution of the initial-value problem Step 2: lim sup t→+∞ ξ 1 (t) + y(t) . For any given > 0, ∃T 1 > 0 such that ξ 1 (t) ≤ K 1 + 2 for all t ≥ T 1 , as a consequence of Step 1. Furthermore, by the same rationale as in Step 2 of Part 1, we have The associated characteristic polynomial is P l ξ1y = P l(1) ξ1y P l(2) ξ1y , where Since all coefficients in the second degree polynomial P l ξ1y have the same sign, its roots are negative.Thus, E l ξ1y is LAS iff the root of P l(1) ξ1y is negative, which is equivalent to C.2. Non-coexistence equilibrium states of S o .The Jacobian to be used for evaluating the local stability properties of the considered equilibrium states, in this case, is obtained considering the first three rows/columns of J in ( 22) and imposing Thus, E o ξ1 is unstable by the non-triviality assumptions (5).The LAS subspace is (ξ 1 , 0, 0).

E
1−e yξ 1 ezy 0 0 (q z + C) g yξ1 Consequently, from the definition of P o(1) ξ1y , the following condition is necessary and sufficient for the considered equilibrium to be LAS: In this case, the associated characteristic polynomial is ξ1z , where Since all terms in P o ξ1z have the same sign, its roots are negative.Thus, in order to guarantee that E o ξ1z is LAS, we have the impose that the root of P o(1) ξ1z is negative, which is true iff e zξ1 e yξ1 − 1 C.3.Non-coexistence equilibrium states of S p .In this case, the Jacobian is obtained from the general Jacobian matrix ( 22) by assuming D zξ1 = 0.

Figure 1 .
Figure 1.The model of the trophic chains under analysis including the welfare compartment; arrows represent biomass transfers.

Figure 2 .
Figure 2. Types of trophic chains under analysis.

Figure 4 .Part 1 .
Figure 4. System S p : optimal steady states and control inputs as functions of δ.