Backward bifurcation, oscillations and chaos in an eco-epidemiological model with fear effect

This paper considers an eco-epidemiological model with disease in the prey population. The disease in the prey divides the total prey population into two subclasses, susceptible prey and infected prey. The model also incorporates fear of predator that reduces the growth rate of the prey population. Furthermore, fear of predator lowers the activity of the prey population, which reduces the disease transmission. The model is well-posed with bounded solutions. It has an extinction equilibrium, susceptible prey equilibrium, susceptible prey-predator equilibrium, and coexistence equilibria. Conditions for local stability of equilibria are established. The model exhibits fear- induced backward bifurcation and bistability. Extensive numerical simulations show the presence of oscillations and occurrence of chaos due to fear induced lower disease transmission in the prey population.


Introduction
Predator-prey interactions are a central topic of discussion in studying ecological communities. These interactions are often altered in nature due to the presence of infectious disease that affects the prey, the predator, or both. Understanding the predator-prey-pathogen dynamics requires the development and analysis of population models where one or more of the main populations are subjected to an infection. Models that incorporate disease in ecological communities are called eco-epidemiological models, and represent a natural extension of more classical population interaction models [35]. The first eco-epidemiological model with disease in the prey was introduced by Anderson and May [1]. This early model was followed by the work of a number of researchers in eco-epidemiology [2,4,9,15,16]. Eco-epidemiology now is a branch of mathematical biology connecting ecology with epidemiology. Although the literature of eco-epidemiology is rich enough, the impact of fear of predators on eco-epidemiological systems is not properly studied yet. Recent experimental results revealed that fear of predators can reduce the growth rate of prey population and change the foraging behaviour [32,39]. In general, basal prey lower their predatory activity in the presence of their predators. According to optimal foraging theory, prey increase their survival probability by avoiding high-risk grazing zone and remain starving or grazing on the lower intake zone [6]. Such lower foraging activity reduces the chance of contacts between infected and susceptible individuals. Therefore, fear of predator can have a great impact on the dynamics of eco-epidemiological systems.
In classical predator-prey theory, the impact of predators on the prey population has been described through only direct killing of prey by predators, because the effect of direct killing on prey numbers can be directly observed. However, an emerging view is that the indirect effect of predator on prey numbers may be far greater than direct predation. Due to predation risk, all prey show a variety of anti-predator responses such as changes in foraging behaviours, changes in habitat usage, etc. [11]. Zanette et al. [39] have performed experimental studies and showed that predator-prey population dynamics has been affected enormously by the cost of fear. They eliminated direct predation by protecting every nest in the experiment with both electric fencing and seine netting and began predator play back several weeks before the first egg of the season was laid and continued predator call and sound broadcasts throughout the 130-days breeding season. They observed the reduction in numbers of eggs, hatchlings, and fledglings in the successive generations. They observed that the number of offspring was reduced by 40%. They demonstrated that the prey's perception of predation risk alone is powerful enough to affect the population growth rate [39]. Other evidence suggest that fear can affect populations like snowshoe hares [28] and dugongs [37]. Recently, another field experiment performed by Suraci et al. [32] demonstrated that the fear of large carnivores can provide significant service in conserving the ecosystem function. In their experiment, they manipulated fear using month-long playbacks of large carnivore vocalizations. The experiment was conducted on wild, free-living mesocarnivores on several small coastal Gulf Islands. As a result, they showed that fear of large carnivore reduced mesocarnivore forging and increased vigilance. They also observed mesocarnivore's prey (interdial crabs, interdial fish, polychaete worms, subtidal red rocks crabs) were benefited due to fear of large carnivores. The authors concluded that the lower trophic level (prey) is benefited from the fear among mesocarnivores, and it could be useful in ecosystem conservation.
Recently, Wang et al. [36] modified the Rosenzweig-Macarthur predator-prey model [23] by considering the cost of fear in prey. Incorporating the cost of fear, the authors obtained both supercritical and subcritical Hopf bifurcation which is in contrast with classical predator-prey models where Hopf bifurcation only be supercritical. The results also showed that high levels of fear stabilize the predator-prey system by excluding the existence of periodic solutions.
On the other hand, during epidemic outbreak fear of infection can shape the force of infection by inducing behavioural changes and ultimately reduce disease prevalence. Capasso and Serio [7] studied an SIR model, where they considered the saturation phenomena for large numbers of infectives to capture the psychological effects. Recently, Epstein et al. [14] studied an SIR model with fear. Individuals with fear are assumed to respond with only two actions, namely self-isolation and spatial flight [14]. The authors studied spatial flight as a behavioural response and concluded that small levels of fear-inspired flight can dramatically reduce the epidemic size. Although the impact of fear in the epidemic outbreak has been studied extensively, however, there is lack of under standing how the fear of predators can change the dynamics of eco-epidemiological systems.
In the present paper, we have considered an eco-epidemiological model with disease in the prey population. We assume that fear of predator reduces the reproduction rate [36] of the prey population. We also assume that fear of predator lowers the foraging activity and increases the vigilance in the prey population [32], which consequently reduces the probability of getting infected. Lower activity in prey implies lower chance of contact between susceptible and infected prey populations. In the present work, we study an ecoepidemiological model where prey population is subjected to disease infection and fear can reduce prey growth rate, lower foraging activity, which consequently lowers the force of infection.
The rest of paper is organized as follows: In Section 2, we formulate the model in which the prey population responds behaviourally with fear to perceived predation risks. In Section 3, we check the positivity and boundedness of solutions of our model. In Section 4, we determine the equilibria and analyze our model with fear. In Section 5, we establish the presence of backward bifurcation for the predator-free equilibrium and determine rigorously the direction of the bifurcation. In Section 6, we perform some numerical simulations which reveal that the fear effect plays a crucial role in eco-epidemiology. At the end of this paper, we discuss the biological significance of our mathematical results and conclusions.

Mathematical formulation
Here we consider a predator-prey system, where the prey population is subjected to infection. Let u(t) be the prey population density and v(t) be the predator population density at time t. We consider that birth rate of prey is r 0 , d is the natural death rate of prey and a represents the death rate due to intra-species competition. To incorporate the fear phenomena, we multiply the reproduction term i.e. birth rate (r 0 ) of susceptible individuals with a decreasing function of the predator population size, f (k 1 , v) = 1/(1 + k 1 v), suggested by Wang et al. [36]. Here k 1 be the level of fear that reduces the growth rate of susceptible prey. From the biological point of view, f (k 1 , v) is appropriate since Next we divide prey population into two subclasses, susceptible prey (u 1 ) and infected prey (u 2 ). We assume that only susceptible prey can reproduce and the disease is not genetically inherited. We also assume that infected prey do not compete for the resource for being weak due to disease infection. The susceptible prey becomes infected only through a contact with the infectious prey at a rate β. We model the incidence through mass action law βu 1 u 2 . We consider that fear of predator reduces the foraging activity among prey [32], which in turn reduces the disease transmission rate. We assume that scared prey spreads the disease at a rate β/(1 + k 2 v), where k 2 be the cost of fear that lowers disease transmission.
In predator-prey theory, choice of a predator functional response is very crucial for modelling predator-prey dynamics when prey is divided into susceptible and infected compartments. Researchers [21,24,26] considered Holling type II functional responses for multiple prey populations, where all prey populations contribute to the saturation. Therefore, the functional responses are given by f 1 (u 1 ) = p 1 u 1 /(1 + q 1 u 1 + q 2 u 2 ) and f 2 (u 2 ) = p 2 u 2 /(1 + q 1 u 1 + q 2 u 2 ), where both the susceptible and infected prey contribute to saturation. Here p 1 and p 2 are the predator's attack rate on susceptible and infected prey and 1/q 1 and 1/q 2 are the half-saturation constants, respectively.
From the above assumptions we obtain the following system of nonlinear differential equations: where, c 1 and c 2 are the conversion efficiencies of captured susceptible and infected prey into predator biomass. μ and m are the death rates of infected prey and predator populations, respectively. The system has to be analysed with the initial conditions u 1 (0) ≥ 0, u 2 (0) ≥ 0, v(0) ≥ 0 and all parameters are assumed non-negative. We show below that the model is mathematically well posed in the positively invariant region and solutions in X exist for all positive time.

Positivity and boundedness of solutions
, according to ecological significance. The right-hand side of system (1) is continuously differentiable and locally Lipschitz in the first quadrant X = {(u 1 , u 2 , v) : u 1 , u 2 , v ≥ 0}. Therefore, Theorem A.4 in [33] implies that the solutions of the initial value problems with nonnegative initial conditions exist on the interval [0, b)[0, ∞) Lemma 3.1: If c 1 < 1 and c 2 < 1, then the solutions of system (1) that start from initial conditions in R 3 + eventually enter the region S defined by the set S = {(u 1 , u 2 , v) ∈ R 3 + |F = δ+ H/N for some δ > 0}.

Proof:
We define a function F as The derivative of (2), with respect to time iṡ where the dot represents the derivative with respect to time. Assume c 1 < 1 and c 2 < 1. Then we haveḞ Now we choose an arbitrary positive real number N for whichḞ Thus, for large values of t we have 0 < F ≤ H/N. Consequently, solutions of the system that are initiating in R 3 + eventually lie in the region S defined by

Equilibrium analysis
System (1) admits the following five non-negative equilibria.
(ii) The disease-free and predator-free prey equilibrium if the reproduction rate of susceptible prey is larger than the death rate of susceptible prey. Under the condition r 0 > d, E 1 always exists. (iii) The predator-free equilibrium E 2 = (ǔ 1 ,ǔ 2 , 0), whereǔ 1 if the intrinsic growth rate of susceptible prey is larger than a threshold value determined by the ratio of death rate of infected prey and the disease transmission rate. (iv) The disease-free predator-prey equilibrium andv is the root of the given equation, This quadratic equation has a unique solutionv > 0 iff r and c 1 p 1 > mq 1 , i.e. if the intrinsic growth rate of susceptible prey is larger than the sum of the death rate of susceptible prey and density dependent death rate a times the susceptible prey at equilibriumū 1 .
is a positive solution of the system of equations To prove its existence we define the following reproduction numbers: First, we define predator invasion number of the infectious equilibrium of the prey as Second, we define disease invasion number of the predator-prey disease-free equilibrium
Then there exists at least one coexistence equilibrium E * .
From third equation in system (3) we can express u 1 as a function of u 2 . This follows from the implicit function theorem. Therefore, where f is a continuous and increasing function for all u 2 ≥ 0 whereū 1 = f (0) and exists for all u 2 ≥ 0. Since c 1 p 1 − q 1 m > 0, we have assumed for simplicity Therefore, predator can persist on u 1 alone but not on u 2 alone. From the first equation of system (3), we express v as a function of u 2 , say v = G(u 2 ). This equation may have more than one solution for v as this equation is not monotone in v. We rewrite the first equation in the form We neglect the second case and assume u 1 , u 2 are such that dG/du 2 is defined and positive for all u 2 such that Assume u 2 = 0, then where G(0) is the solution in v of the equation The solution of the above equation isv. Hence,v = G(0), we have as we assume that R 0 > 1. Now let u 2 = u * 2 and we get Further, v = 0 is also a solution of this above equation, i.e. Therefore, We know that There are two possibilities occur, either Here, Case-(B) is impossible as from Equation (4) we have,ǔ 1 > f (ǔ 2 ). Now using Case-(B), we have This concludes the proof.

Local stability analysis
To study the local stability properties of the equilibrium points, we calculate the Jacobian matrix around each equilibrium point of system (1). The Jacobian matrix of the model at where The following lemmas show the local stability of equilibria.
The predator-free equilibrium point E 2 is locally asymptotically stable if R p < 1.
For the proof of this Lemma 4.1(iii) see Appendix 1.
Proof of this lemma is given in Appendix 2.
Proof of this lemma and supportive calculations are given in Appendix 3.

Hopf bifurcation analysis
In this section, we have explored the possibility for occurrence of Hopf bifurcation of system (1). We observe that system (1) undergoes a Hopf bifurcation for gradual increase of the disease transmission rate. Hopf bifurcation around the interior equilibrium E * of system (1) with bifurcation parameter β is given in the Theorem below.

Theorem 4.2:
When the disease transmission rate (β) crosses a critical value, system (1) exhibits Hopf bifurcation around the positive coexistence equilibrium. The necessary and sufficient condition for Hopf bifurcation [22] to occur is that there exists β = β c such that, where λ is the root of the characteristic equation corresponding to the interior equilibrium point.
Proof: For β = β c , we can write the characteristic equation For all β, the roots are in general of the form Now, we shall verify the transversality condition d dβ (Re(λ(β))) | β=β c = 0, j = 1, 2.
Substituting λ j (β) = η 1 (β) + iη 2 (β) into the characteristic equation and calculating the derivative, we have where, Therefore, the transversality condition holds. This implies that a Hopf bifurcation occurs at β = β c . This concludes the proof of the Theorem.

Backward bifurcation analysis
Recently, Boldin [5] investigated the possible effects of an invasion when the parameters of a model are varied so that R 0 of the invading population passes the value 1. They performed uniform study of ecological, adaptive dynamics and disease transmission models and derived a simple formula for the direction of bifurcation from a steady state in which only the resident populations are present. In the present investigation, we also investigate the backward bifurcation and the direction of backward bifurcation for our model. Theorem 4.1 shows that a coexistence equilibrium exists. Here we show that the coexistence equilibrium may not be unique. Co-existence equilibria in this system can occur in one of two ways: (1) if predator invades prey-disease equilibrium; (2) if disease invades predator-prey equilibrium.
Here we show that fear in the prey generates backward bifurcation and allows the predator to persist for R p < 1. To eliminate the effect of the Holling II functional response terms, we take q 1 , q 2 = 0. Thus system (1) for E * becomes From the above system of equations we get, Therefore, c 1 p 1 f 1 (v) + c 2 p 2 f 2 (v) = m, that is, v is a solution of the following equation: Plotting the solution of the above equation in the (R p , v) plane for gradual increase of c 1 . We see that the curve bifurcates backward from the critical value c * 1 such that R p (c * 1 ) = 1 (see Figure 11). Plotting with respect to R p (rather than c 1 ) in Figure 11, we see that even if R p < 1 the predator may still persist. Therefore, the invader (predator population densities) can meet with success even if R p < 1 [5]. Thus fear in prey allows the predator to persist for values of its invasion number below one.

Direction of backward bifurcation analysis
When R p is less than unity, system (1) shows backward bifurcation i.e. a small positive unstable coexistence equilibrium appears while the predator-free equilibrium and a larger positive coexistence equilibrium are locally asymptotically stable. For the clarification, we outline a theory which determines the backward bifurcation of the predator-free equilibrium. This theory is based on the general centre manifold theory [8,12,18].
Here we consider a system of autonomous differential equations where φ is a bifurcation parameter and u = (u 1 , u 2 , u 3 , . . . · · · , u n ) ∈ n . Without loss of generality, it is assumed that0 = (0, 0, 0, . . . ., 0) n-times is an equilibrium for system (5) for all values of the parameter φ, that is Now we state a particular part of Theorem (4.1) of [8] and give a rigorous proof for the backward bifurcation of our system (1).
Also let,â > 0,b > 0. When φ < 0 with |φ| 1,0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when 0 < φ 1,0 is unstable and there exists a negative and locally asymptotically stable equilibrium and at φ = 0 a backward bifurcation takes place.
For system (1), let φ = c 1 be the bifurcation parameter. When R p = 1, we considered the bifurcation point as where we have used the predator-free equilibrium The linearization matrix of system (1) around the predator-free equilibrium point when φ = φ * is It is easy to see that 0 is an eigenvalue of D u (f ). A right eigenvector associated with the 0 eigenvalue isw = (w 1 , w 2 , w 3 ) t , where A left eigenvectorh associated with the 0 eigenvalue, satisfyingh ·w = 1 ish = [0,0,1]. Furthermore, at the predator-free equilibrium point we get, The rest of the second derivatives appearing in the formula forâ andb are all zero. Hence, Therefore, if So, the direction of the bifurcation of system (1) is backward for k 2 > k 2 . Figures 11(a,b) show that there is a parameter set for which these conditions may occur.

Numerical results
In this section, we have performed some numerical simulations on system (1) to illustrate the analytical results observed in the previous sections. All the simulations are carried out in MATLAB. This study demonstrates stability, the presence of limit cycle, Hopf bifurcation, bistability, higher order periodic oscillations and chaos. Here disease transmission rate β and cost of fear k 1 , k 2 are important parameters under investigation. For simplicity, we consider k 1 = k 2 = k throughout the simulation. For the above set of parameter values, the system shows that the endemic coexistence steady state is stable (see Figure 1). In Figure 1, we see that the trajectories starting inside the region of attraction approach the endemic coexistence equilibrium point (E * ).
To investigate the impact of fear, we fix β = 0.381 and gradually increase the strength of fear (k). For k = 0.5, we observe that the disease becomes extinct from the system and susceptible prey and predator populations coexist in a stable manner (Figure 2).
To observe the long-term behaviour of system (1) for a range of values of β, we draw the bifurcation diagram considering β as a bifurcation parameter. In Figure 4, we observe that the system shows limit cycle oscillations for 0.3987 < β ≤ 0.3999; Hopf bifurcation (HB) occurs at β = 0.39999 and system become stable focus for 0.3999 < β ≤ 0.4012, and for β > 0.4012, the predator population becomes extinct through transcritical bifurcation (TB) from system (1) while susceptible and infected prey populations co-exist in a stable mode.
To investigate the impact of fear when the system is unstable around the interior equilibrium, we fix the value of β at 0.399 such that susceptible prey, infected prey and predator populations show periodic oscillation. We draw the bifurcation diagram with respect to k (see Figure 5). In Figure 5, we observe that if we gradually increase k then above a threshold value (k = 0.5) the system enters into chaotic regime from limit cycle oscillations. If we further increase k, then above a threshold value (k = 6) the infected population goes to extinction while the susceptible prey and predator populations coexist in a oscillatory manner. Then above a critical value k = 12.75, the system becomes stable around the disease-free predator-prey equilibrium E 3 .
For k = 4, system (1) shows chaotic behaviour (see Figure 6), where other parameter values are as same as in Figure 3. In Figure 7, we draw the Poincare map of system (1) for β = 0.399, k = 4 and rest of the parameter values are as same as in Figure 3. Here we fix respectively. So here we get a bistability. Bistability is a phenomenon where the system converges to two different equilibria for the same parameter values based on the variation of initial conditions. Here any trajectory starting from the interior R 3 + either converges to E * or E 2 . To find the basin of attraction for the bistability, we plot Figure 10, where the blue dotted region is the basin of attraction for the equilibrium point E * and the red dotted region is the basin of attraction for the equilibrium point E 2 . In Figure 10   plane; in Figure 10(b), for initial density of u 2 at 0.02095 we get the basin of attraction of system (1) on the u 1 − v plane.
Further, to observe the backward bifurcation of system (1) with respect to the predator invasion number R p between the predator-free equilibrium and the coexistence equilibrium, we plot the backward bifurcation diagram (see Figure 11) for the corresponding set of parameter values: β = 0.4, r 0 = 0.5, k 1 = 25, d = 0.01, a = 0.01, p 1 = 0.5, q 1 = 0, m = 0.35, p 2 = 0.5, q 2 = 0, k 2 = 250, c 2 = 0.501, μ = 0.08. In Figure 11, we take R p as a bifurcation parameter. We vary the parameter c 1 in the region [0.33, 0.35] and plot R p along x-axis and plot the equilibrium predator density along y-axis. Figure clearly shows that fear permits the predator to exist for values of its invasion number below one.
In Figure 11, the invasion curve marked by red and blue colour is a curve of backward transcritical bifurcation of equilibrium densities [19]. Here we draw blue curve and black line for stable branches of the endemic equilibrium and predator-free equilibrium, respectively. The red curve and green line indicate the unstable branches of the endemic equilibrium and the predator-free equilibrium, respectively. Here for predator invasion number R p < 1, if we take any point from the area between the red curve and black line, then an unstable interior equilibrium appears and the predator-free equilibrium becomes stable. Again, for R p < 1, if we take any point from the area between the blue and red curve, then a stable interior equilibrium also appears. Therefore, backward bifurcation creates a bistability between endemic equilibrium (E * ) and predator-free equilibrium (E 2 ). Further, we choose c 1 = 0.339 when other parameter values are fixed as in     Figure 11 such that the predator invasion number is less that unity (R p = 0.97). For this set of parameter values, we compute the interior equilibrium points. For above parameter values, system (1) has two endemic equilibrium points E * (8.2889, 0.1376, 0.0986) and E * (1.6564, 4.4866, 0.0247), where first one is locally stable and the later is unstable.

Conclusion
In this paper, we have considered the scenario that fear of predator reduces the reproduction rate of prey population and it also suppresses the disease transmission among the prey population. Several studies find reduced reproduction in species due to fear of predator [3,27,31]. Fear effects on any population can measure the influence of long-term population dynamics and ecosystem function [13]. Not only fear changes the ecosystem function but also influences epidemic functions. So to make the model more biologically relevant to the population dynamics, incorporation of fear effect in the model is an important factor.
We have explored a large variety of complex dynamics, which is much broader than the previous studies in ecology and eco-epidemiology. In the present study, we explore the rich dynamics of the eco-epidemiological system for Holling type II functional response, when both susceptible and infected prey contribute for saturation.
To study the impact of fear in the eco-epidemic system, we consider two situations where the eco-epidemiological system shows stable dynamics and unstable (limit cycle oscillations) dynamics. Then we gradually increase the strength of the fear and explore the impact of fear in an eco-epidemic system. We observe that when the system is stable around the endemic equilibrium, if we increase the strength of fear, then the disease will be wiped out from the system. We also observe that when the system shows limit cycle oscillations around the endemic equilibrium, if we increase the strength of the fear, then above a critical value of fear, the disease will become extinct from the system and the system becomes purely ecological system. However, for intermediate value of fear, the system shows chaotic oscillation. Therefore, fear in the eco-epidemiological systems may produce chaos. Chaotic oscillations may potentially be explained by the incommensurate frequencies of population cycles. If the frequency of the oscillation of the susceptible-infected prey subsystem is incommensurate with the frequency of the oscillation in either susceptible prey-predator subsystem or infected prey-predator subsystem, then the system may produce chaos [17]. If we further increase the strength of the fear, then the system becomes stable, replacing the population oscillations by a stable disease-free predator-prey equilibrium. Therefore, if the level of fear increases, then the infected prey population goes to extinction from the endemic steady state or oscillations of coexistence. Fear in prey population lowers the growth rate and the foraging activity. Lower foraging activity reduces the chance of being infected. Therefore, in the presence of fear, the effectual rate of disease transmission reduces significantly, which may lead to eradication of disease from the system. We observe that fear can suppress infection in both situations, stable endemic state and coexisting oscillations.
We further explore some rich dynamics such as Hopf bifurcation and bistability in our model. We observe that if the disease transmission rate is increased gradually, then the predator-prey oscillations become stable via Hopf bifurcation. We observe that disease can stabilize the population oscillation by replacing limit cycle oscillation by stable coexistence equilibrium. As the disease transmission rate increases more prey individuals become infected which also contribute for the saturation on predator consumption on susceptible prey. Such indirect saturation effect may lower the consumption of susceptible prey and enhance the stability of the eco-epidemic system. For a different set of parameter values, we observe a bistability between predator-free equilibrium (E 2 ) and endemic equilibrium (E * ). Depending on the initial condition the system converges to either predator-free or endemic steady state. In contrast, Siekmann et al. [29] found a bistability between diseasefree and predator-free equilibrium in a predator-prey system with disease in prey. The bistability in our system is likely produced by a backward bifurcation of the coexistence equilibrium with respect to the predator invasion number. In bistability situation, predator population may become extinct from the system, where initial population density plays a crucial role in the persistence of the predator population. Fear is a necessary factor for the backward bifurcation and allows for the predator to persist even if its invasion number is smaller than one. Therefore, bistability is an important issue as it relates to the predator persistence and extinction.
Several studies in eco-epidemiology describe complex dynamics. To the best of our knowledge, the first eco-epidemiological paper to show chaos is Upadhyay et al. [34] who used an existing model of Chattopadhyay and Bairagi [10]. They showed the chaos via a cascade of period-doubling bifurcations. Stiefs et al. [30] described quasi-periodicity and chaos through a generalized predator-prey model with disease in the predator population. Siekman et al. [29] found bistability in a predator-prey system by incorporating a free-living virus stage in the model and a disease in the prey population. Kooi et al. [20] found more complex dynamics, including period-doubling cascade into chaos, bistability and transcritical bifurcations of limit cycles in an eco-epidemiological system. Recently, Saifuddin et al. [25] also demonstrated that disease can produce chaos in a predator-prey model. However, the present investigation first time reports that fear factor has the potential to produce backward bifurcation and chaos by suppressing prey growth and disease transmission.