INTRODUCTION

Until recently, models of sustainability in the tourism literature have commonly lacked systematic grounding in concepts of sustainable development, reflecting a broader lack of formalism in common discussions of sustainability (Chichilinsky, 1997; Collins, 1999; Johnston and Tyrrell, 2005). A small number of works, however, now offer more formal, mathematical models of tourism sustainability. These include the models of Casagrandi and Rindaldi (2002) and Johnston and Tyrrell (2005), both of which address tourism sustainability within a dynamic optimisation framework. One of the primary factors distinguishing these works from others is a basis in formal mathematical (or quantitative) models; this stands in contrast to qualitative or case study approaches that characterise most of the sustainability literature ( Johnston and Tyrrell, 2005). Although some might view such models as little more than complex mathematical abstractions, the abstract, formal nature of these models allows for the provision of insights often unavailable through alternative approaches (Casagrandi and Rinaldi, 2002; Johnston and Tyrrell, 2005).

As stated by Henderson and Quandt (1971: 2): 'theories represent simplifications and generalizations of reality and therefore do not completely describe particular situations... theories are fruitful because they contain statements which abstract from particulars and find elements which many situations have in common. Increased understanding is realized at the cost of sacrificed detail'. Such tools should not be considered 'operational tools for managers', but rather mechanisms through which one may explore more universal patterns related to sustainable and non-sustainable outcomes (Casagrandi and Rinaldi, 2002: 14). Moreover, the quantitative formalism of such models is such that results often follow directly and inescapably from model assumptions and structure. The concomitant potential of these models to generate unexpected outcomes — or results that sometimes challenge 'common wisdom'— renders them particularly well suited to the stimulation of debate into appropriate paths to tourism sustainability.

The potential advantages of models such as those of Johnston and Tyrrell (2005) and Casagrandi and Rindaldi (2002) are perhaps most easily understood when the results of such models are contrasted to ambiguous definitions of sustainability that pervade policy dialogs. An example is the commonly cited definition of sustainability found in the Brundtland Report: 'development that meets the needs of the present without compromising the ability of future generations to meet their own needs' (World Commission on Environment and Development, 1987). As highlighted by Johnston and Tyrrell (2005), such definitions assume relatively undisputed social goals and an ability to agree on policies that meet, for example, the needs of the present. These assumptions notwithstanding, there is rarely perfect agreement on those policies that are most appropriate for satisfying even 'the needs of the present', much less those that satisfy the complete Brundtland definition. Formal models can highlight limitations in standard approaches to (and definitions of) sustainability and illuminate trade-offs that are fundamental to sustainability in various tourism contexts.

Johnston and Tyrrell (2005), for example, develop a mathematical model illustrating the interrelated dynamics of tourism-related economic and environmental conditions over time. Using insights from the model, they demonstrate that operational definitions of tourism sustainability require details regarding elements that are to be sustained, the users for whom they are to be sustained, and the levels at which they are to be sustained. Absent such critical details, the concept of sustainability remains more of a guiding fiction than an operational concept — devoid of the substance needed to identify necessary policy trade-offs.

Johnston and Tyrrell (2005) also emphasise that there are many different groups affected by tourism, and that these groups will likely realise different benefits and costs from otherwise identical types of tourist activities. Their model demonstrates that while there are many different outcomes and paths that may be consistent with sustainable tourism, there are likely none that will be considered optimal by all affected groups. For example, one may seek to sustain the number of visitors, the size or growth of industry profits, the quality of some or all environmental resources, the quality of the tourist experience, the number of tourist jobs, the quality of life of local residents, or some combination of these and other elements. It is unlikely that all may be sustained simultaneously, nor that the particular variant of sustainability most preferred by one user group will be favoured by others.

This paper uses insights from the basic model of Johnston and Tyrrell (2005) to frame the study of sustainability in tourism planning, with a particular focus on the role of dynamic resilience. Specifically, we suggest that the two basic equations of Johnston and Tyrrell's (2005) model identify two areas of research that will underlie practical attempts to implement policy for sustainable tourism. We also identify potential extensions of the model to introduce the concept of resilience to the sustainable tourism dialog — or the ability of social, economic or ecological systems to recover from tourism-induced stress. The specific goals of the paper are to characterise a more quantitative approach towards tourism sustainability and resilience, to summarise results in the tourism literature that might contribute to initial empirical implementation of such quantitative models, and to stimulate debate regarding the potential role of such approaches in guiding tourism policy.

DYNAMIC MODELS OF TOURISM SUSTAINABILITY — A QUICK REVIEW

This section provides a brief review of the basic model of Johnston and Tyrrell (2005), as a foundation for subsequent discussion. For simplicity, we consider only the version of the model for the tourism industry (the original article contrasts this with a model for residents of a tourist destination). The model contains two basic structural equations: one that characterises the long-term industry objective (long term, or sustainable profits) and another that describes unavoidable dynamic trade-offs in the system. As implied above, the model maintains a relatively high degree of abstraction to preserve generality and simplify analysis. The fundamental trade-offs in the model relate to the dynamic interplay among visitor numbers, environmental quality, and profits.

The first equation characterises industry profits as determined by the combination of visitor numbers and environmental quality. Profits increase as the number of visitors increases, holding all else equal, but each new (additional) visitor adds a little less to industry profits on the margin. Similarly, profits increase as environmental quality increases, holding all else constant, because visitors will pay more to visit higher-quality locations. Each successive improvement to environmental quality, however, results in a slightly smaller gain in industry profits, again reflecting diminishing marginal returns.

Johnston and Tyrrell (2005) formalise this relationship with the following equation, in which the tourism industry chooses the number of visitors (V) in each time period to maximise the sum of discounted profits over time:

In equation (1), environmental quality is given by X, profits are given by the general mathematical function (V, X), and the integral reflects the fact that industry's goal is the maximisation of the discounted sum of profits over time. Additional details of this equation, which follows standard approaches in the economics literature, are provided by Johnston and Tyrrell (2005).

The second structural equation characterises the dynamic relationship between tourist activity and environmental quality. This second equation is the primary focus of this paper. To maintain simplicity, the original model maintains a general definition of 'environmental quality'. Tourists both are attracted by and degrade environmental quality, which in turn is renewed based on natural processes. The equation describing the total per-period change in environmental quality combines the negative influence of visitors V and the positive influence of a natural growth/renewal function h(X):

In equation (2), indicates the change in environmental quality from period to period, which may be positive, negative, or zero. Sustainability of environmental quality occurs where =0, that is, there is no change in environmental quality over time (  Johnston and Tyrrell, 2005). This implies finding one of the balance points between natural renewal and visitor damage.¹

Although there may be many potential points that balance natural renewal and visitor damage, the question for tourism planning is how to choose the specific point(s) that both maintain this sustainable balance and also maximise long-term discounted net benefits, in this case to the tourism industry.² The calculus of variations provides a solution for this problem, as illustrated by Johnston and Tyrrell (2005). Aside from identifying this solution, Johnston and Tyrrell (2005) discuss implications for trade-offs in tourism sustainability and for both short- and long-term tourism planning.

RESILIENCE — THE CONCEPT

This paper extends the basic model of Johnston and Tyrrell (2005) to introduce the concept of resilience or resiliency to the sustainable tourism dialog. While definitions of resiliency differ across disciplines, we believe that the term may be used to accurately characterise — in a general sense — the dynamics of community environmental quality (or character) subject to visitor-induced degradation. This is the relationship captured in general form by equation (2) above. We have considered alternatives such as 'adaptability' and 'recovery' to describe this equation, but return to 'resilience' primarily because of its popularity in health and community studies. Like the word 'sustainability', which is also poorly defined, the term 'resilience' will undoubtedly take on different and expanded meanings as it is used by practitioners and other disciplines.

In physics and engineering, for example, resilience is defined as the capacity of a material to absorb energy when it is deformed elastically and upon unloading to have this energy recovered. In other words, it is the maximum energy per volume that can be stored. In the context of systems ecology, one of the most familiar definitions is provided by the Resilience Alliance research consortium, which defines resilience as 'the capacity of an ecosystem to tolerate disturbance without collapsing into a qualitatively different state that is controlled by a different set of processes' (Brozovi and Schlenker, 2007: 2). That is, resilience is related to a disturbance threshold beyond which a system collapses or shifts into a fundamentally different state (Johnston and Sutinen, 1996).

The concept of resiliency may be applied to numerous entities, defined at different levels of aggregation. For example, in the context of organisational resiliency, Carnegie Mellon (2007) defines operational resiliency as '... the organization's ability to adapt to and manage risks that emanate from day-to-day operations. Organizations that have resilient operations are able to systematically and transparently cope with disruptive events so that the overall ability of the organization to meet its mission is not affected'. In personal health '[r]esiliency is the ability to spring back from and successfully adapt to adversity. An increasing body of research from the fields of psychology, psychiatry, and sociology is showing that most people — including young people — can bounce back from risks, stress, crises, and trauma and experience life success' (Henderson et al., 1999). Within such individual-focused definitions, 'resiliency has been described as the capacity for successful adaptation, positive functioning or competence despite high risk, chronic stress, or following prolonged or severe trauma' (Egeland et al., 1993; Cowen, 1991). One may also apply the concept to families or communities (McCubbin and McCubbin, 1993; Health Canada, 2007).

Resilience and tourism dynamics

In a dynamic model of tourism such as that of Johnston and Tyrrell (2005), uncertainty and environmental resilience may be incorporated as an uncertain 'resilience threshold' in environmental quality, below which tourism shifts or collapses to a fundamentally different state. For example, below a certain threshold of environmental quality an ecosystem may collapse such that tourism is no longer viable. Within this context, we define environmental quality broadly to capture the overall quality of the tourist environment, whether it be an urban, rural or other setting. In an urban tourism setting, for example, declines in site attractiveness³ beyond some threshold may cause an irreversible shift from 'luxury' to 'budget' visitors (or to no visitors at all), with a concomitant change in viable businesses. Similarly, in an ecotourism case, certain resources upon which tourism depends (eg rare or sensitive wildlife species) may disappear if a certain level of tourist impact is reached. In these and other cases, the relevant thresholds are likely uncertain — tourism planners may not know with certainty the resilience threshold beyond which the generally irreversible system change will occur. The formal mathematics of uncertain resilience and ecosystem shifts can be complex and are contained in Johnston and Tyrrell (2007).

RESILIENCY — EXTENDING THE MODEL

In the second equation of the dynamic model of tourism there are only two variables: the number of tourists and environmental quality. It is assumed that the primary variable over which planners have control is tourist numbers. This choice, in turn, influences environmental quality, which subsequently affects the ability of the area to attract higher-paying tourists. This, of course, is a simplification. In reality, a wide variety of variables influence the quality of a tourist destination, and 'environmental quality' has a number of important dimensions. These dimensions⁴ are highlighted in the tourism literature, but not combined in a single, formal dynamic framework.

Here, we propose a simplified structure for different possible dimensions of environmental quality, broadly defined. This is done at risk of considerable oversimplification of the situation facing real-world tourism destinations, but also allows a formal mathematical model to be developed and explored. This simplified model structure requires four basic assumptions. First, we assume that tourist destination quality may be represented by three dimensions or vectors: ecological-environmental quality (X_n), economic-fiscal quality (X_e) and social-cultural quality (X_c). The academic literature frequently discusses the resilience of tourism destinations in terms of such categories, although they are rarely modelled within a mathematical framework. We also assume that the role of government and other destination management organisations may be represented both by the amount of control (G) exerted over environmental qualities and also the timing (t) of controls used. Thirdly, we assume that the workings of the entire system will depend on the size, resource base and level of infrastructure of the community (s). Finally, we assume that the dynamic system accounts for the distinction and interaction between the above-noted actual quality dimensions as well as the perceptions of these dimensions (x_n, x_e, and x_c) held by the market of potential visitors.

Given these assumptions, one might represent the objective portion of this system (the actual quality dimensions) using the following equations, which extend basic equation (2) above:

In addition, the perceived quality dimensions (the images of the three quality dimensions of the destination) might be given by:

where H represents the amount and type of promotional campaigns, as well as unrelated positive and negative media coverage. Here, there is a distinction between the dynamics that determine actual change in various dimensions of destination quality (2a), and those that determine analogous perceived qualities that directly influence visitor behaviour (2b), for example, the number and types of tourists visiting a given destination. Although equations (2a) and (2b) are mathematically related, the term H allows for publicity and other actions that may drive a wedge between actual and perceived changes in quality.

A final element required to capture a realistic concept of resilience in the above systems is the possibility of system collapse if an uncertain resiliency threshold is breached. To do so, we draw from the approach of Johnston and Sutinen (1996), who model optimal fisheries policy for the case in which collapse in a fishery occurs if stock size is driven down to a critical and unknown threshold. This may be integrated into the system above as a simple and irreversible collapse (eg drop to a level insufficient for tourism) in ecological-environmental quality (X_n), economic-fiscal quality (X_e) and/or social-cultural quality (X_c) if one or more of these dimensions drops below critical and uncertain resilience thresholds Z_n, Z_e, and Z_c, respectively. For example, to incorporate an uncertain quality threshold at which collapse occurs, one might assume that Z_i is a random variable with a probability density function f_i(Z_i), for i=n,e,c. The probability F(X_i) that tourism will not collapse at a given level of quality, X_i, is given by

where f(Zi) is a marginal density function. Treatments of such dynamic systems in a natural resource context are discussed in detail by Johnston and Sutinen (1996).

Accordingly, tourism planners might wish to optimise tourism policies (either by controlling the numbers and types of tourists directly through advertisements and promotions or indirectly through investments in environmental resources) subject to natural rates of change in different dimensions of destination quality (2a), related perceptions of change (2b), and the possibility of collapse if an uncertain threshold is reached (3). The dynamic interplay of these three elements — combined with equations defining the particular goal of tourism planning (eg industry profits, quality of life, etc) — would then be used to characterise paths towards sustainable outcomes and trade-offs implicit in these paths. This combination of (2a), (2b) and (3) would ultimately replace equation (2) in the dynamic model of sustainable tourism. It is likely that mathematical solutions to the dynamic system will be complex and difficult to characterise in general form without additional constraints or assumptions. Nonetheless, if these equations could be estimated, and the dynamic system characterised, for specific tourist destinations they would help solve the complex sustainability problem for a variety of situations.

Unfortunately, the tourism research community is far from being able to formulate or estimate such systems system at any level of realistic detail. However, we do find that literature exists to begin formulating the system. The remainder of this paper discusses some of this literature, with an emphasis on ways in which each contribution might be used to populate the model structure outlined above. We also discuss challenges to potential model implementation. The primary purpose is to summarise patterns that have been found to influence the resilience of tourism in various areas, and to provide greater understanding of the relationship between these empirical findings and the more systematic modelling of tourism resilience introduced above.

Ecological-environmental resilience and tourism

Turton (2005) presents a thorough analysis of the impacts of tourism and recreation activities in natural areas, as they depend on ecological characteristics. He describes resilience as the ecosystem's ability to recover following cessation of visitor activities. 'Regardless of the type of ecosystem, there is a well-documented curvilinear relationship between level of use and the intensity of impact at a site, be it a camping area or a walking track' (p. 145). He contends that few studies examine the recovery of ecosystems following cessation or re-direction of recreation use. Such findings might be used to quantify relationships implied by (2a) or (3) above. Trosper (2002), in contrast, reports ways in which governments can make specific contributions to ecological resilience. Such findings could be used to identify, for example, the role of government control (G_t) in the above dynamic system.

While concepts of environmental resilience are often couched in terms of ecological models and measurable impacts, other approaches are possible. For example, Statistics New Zealand (2007) describes sustainability goals in terms of the relationships between environmental and cultural resilience for native New Zealanders (the Mori): 'In Mori culture, all things have a mauri — a life force. Damage to this mauri, or human attempts to dominate it, result in the mauri losing its energy and vitality. Any loss of mauri affects the lives of people themselves as well as the resilience of ecosystems. Maintaining the mauri of the environment and ecosystem resilience are equally important for sustainable development'. Such concepts, while perhaps of great relevance to tourism planning (eg through impacts on the authenticity of a tourist destination), do not lend themselves to quantification using standard natural science approaches. Hence, even when all dimensions of quality may be identified for a given destination, quantification may remain elusive.

Still other examples show the vital importance of interrelationships between environmental (X_n) and social or cultural resiliency (X_c). Despite limited resources and an underdeveloped tourism infrastructure, the Department of Environmental Affairs and Tourism of the Republic of South Africa specifically focuses on building resilience and adaptive capacity. 'The huge costs and impacts associated with environmental change and its interaction with underlying human vulnerability highlights risk management as a potential key priority for South Africa. Focusing only on improved adaptation to and mitigation of environmental change is not viable in a context where there are multiple stresses, high incidence of poverty, and the high mortality and reduced life expectancy associated with HIV and AIDS. The cumulative evidence for increasing human vulnerability to environmental change calls for a significant policy response and action on several fronts' (Republic of South Africa, 2007). That is, environmental resiliency cannot be maintained in the absence of a resilient social dynamic.

Despite the relevance of such findings to the model outlined above, a challenge to the integration of such findings into general models of tourism sustainability is the case-specific nature of some resiliency patterns. While some general types of resilience may be arguably universal (eg Turton, 2005), others may be place-specific, such that it may be difficult to apply empirical findings from one tourism destination to others. A requirement for individual, place-specific ecological other studies would likely render the application of such models intractable or prohibitively costly for all but high value, high profile tourism destinations.

Economic-fiscal resilience and tourism

The vector X_e above represents economic or fiscal quality — another dimension in which resiliency is critical. Tourism cannot exist without a functioning and viable tourist industry. Numerous endogenous and exogenous factors may influence resiliency in this area. For example, after the terrorist attack of September 11, 2001, travel to many destinations declined dramatically. Nevertheless, Bonham et al. (2006) found that by 2004, gains in Hawaiian tourism receipts from United States visitors more than compensated for the associated losses from Japanese visitors. Economic resilience in this case took the form of substitution of domestic for foreign markets. Other forms of economic resilience may be introduced by a change in tourist attractions. For example, Atlantic City's demise as a nature-based tourist attraction and its recovery based on the 'artificial amenity' of legalised gambling is well documented (Stansfeld, 1978; Wall, 1983). In Atlantic City, the economic resilience of tourism required the replacement of an ecological attraction with an entertainment attraction — the construction of casinos. Such patterns can provide a basis whereby one might begin to identify the elements that might contribute to change in X_e in different types of tourist destinations.

As noted in equation (2a) above, government control can be critical to economic resiliency. For example, after the 1973 Yom Kippur War, Israeli tourism suffered three years of consecutive decline in tourists' arrivals. It then took Israeli tourism three more years to fully recover, largely because the government's response was limited and disorganised (Mansfeld, 1999). In this case, the economic resilience of the tourism industry was influenced by the (lack of) effective government response.

In contrast, effective government response can positively influence the recovery from tourism decline. After the July 1974 Turkish invasion of Cyprus that caused the island's de facto partition into two separate political entities (the Republic of Cyprus in the south and the Turkish occupied north) tourism initially declined. However, between 1977 and 1987 tourist arrivals increased by 18 per cent per year, exceeding the growth rate for other Mediterranean destinations (Ioannides and Apostolopoulos, 1999). In this case the geopolitical separation of political entities positively influenced the economic resilience (ie recovery) of the tourism industry.

In addition to the effectiveness of government response, the size of a state also appears to be a critical feature of economic resiliency. Easter (1999: 403), for example, suggests three main reasons why small states might be potentially more vulnerable than large developing countries: 'lack of diversification, trade dependence and the large impact of natural disasters. Average GDP reflects resilience, suggesting that small states require a higher GDP in order to increase their ability to withstand external shocks without outside assistance'.

We also find evidence that tourist perceptions can strongly influence economic viability. For example, after the 2005 tsunami that devastated Thai tourist destinations, tourists avoided Phuket, Krabi, and Phang Nga. Rittichainuwat (2006) found that even low prices on travel packages could not offset travellers' concerns for personal safety. 'Whereas the Western tourists who went elsewhere were concerned about the incomplete tsunami warning system and disease, the Asian tourists were afraid they would encounter the ghosts of those who were lost'. Here, the perception of the qualities of the destination (ie environmental features related to personal safety) strongly influenced the (lack of) economic resilience. These perceptions, moreover, differed between Western and Asian tourists. In equations (2b) above, such patterns are captured by the functional relationships between the different dimensions of actual and perceived quality at any given destination.

Social — cultural resilience and tourism

Social and cultural resilience may be the most resistant to simple measurement. Nonetheless, it can be critical to the dynamics affecting the actual and perceived authenticity of a destination. As such, it can be as important to tourism sustainability as other types of more easily quantified resiliency. Destinations may have varying degrees of resiliency in this dimension, and government policy may also have a strong impact in this area. For example, Cannery Row in Monterey, California has maintained 'resilience to the standardizing influence of global tourism' through its dedication to historic preservation (Fotsch, 2004). This resistance to cultural change, in turn, has resulted in economic resilience, perhaps possible only because of the large size of the local market, which has supported the niche historic district's survival as a tourist attraction.

As is the case with both environmental and economic resiliency discussed above, social-cultural resiliency is influenced by numerous factors — both within and beyond the control of local tourism planners. For example, social and cultural resiliency — and also relationships between social-cultural and environmental resiliency — likely differ between urban and rural areas. Although the natural ecological systems of urban areas are often hidden or highly degraded and the politics of preservation complex, there are often much larger assets at stake in urban areas and a more compelling need for effective tourism planning. Sustainable policies are generally assumed to be 'too little too late' to make a difference in densely developed urban areas, largely because of the lack of natural vegetative areas. Social and cultural dimensions of urban areas, however, are often subject to substantial change as a result of tourism. Moreover, long-term goals involving the combination of environmental, social-cultural, and economic quality can be even more highly valued by urban residents and downtown tourism properties. Because of the size of populations, numbers of visitors and magnitude of resource use in urban areas, issues of resilience and sustainability in these vectors can be critical to the long-term social well being of a state or region.

Visitor impact on social and cultural resilience may be quite complex, and may not always be negative. A study of visitors to Nepal, Nyaupane et al. (2006), for example, found that residents learned many useful things from tourists, including habits related to personal health and hygiene. Observations from the fieldwork substantiated this notion as health and environmental awareness appeared to be greater in villages engaged in tourism than in areas without tourism. In addition, study participants emphasised that tourism contributed to government recognition of the value of their ethnic cultures as a means to attract tourists to the region. According to the participants, the recognition of the value of diverse ethnic cultures reversed a pre-tourism tendency of governments to assimilate the ethnic groups into the national culture by undermining this diversity. While the complexity of tourism effects on social-cultural resilience may resist simple characterisation in mathematical models, the potential benefits of an increased ability to sustain such qualities are likely substantial for many tourist destinations.

TOWARDS A MORE COMPREHENSIVE PERSPECTIVE ON TOURISM SUSTAINABILITY AND RESILIENCY

The literature on sustainability has thus far focused on short-term policies that reduce human use of natural resources, narrowly defined. Often, research addresses the use and misuse of natural resources in rural areas and pristine reserves where the abundance of natural amenities is obvious and the policies can be implemented by a small number of cooperating government officials and industry leaders. Rarely is the nature of longer-term environmental, economic, social or cultural dynamics studied with regard to the quality of life for residents or destination attractiveness. It is rarer still that the sustainability literature considers the perceptions of tourists as an integral part of resilience dynamics. Nonetheless, there is a considerable literature that addresses many of the individual aspects of sustainability, although most of it is not found under the name of 'sustainability'. While this literature is often qualitative, it provides evidence critical to a more comprehensive perspective on tourism sustainability.

This paper argues for a similar perspective on tourism sustainability that extends beyond narrow perspectives on environmental/ecological resiliency and incorporates critical relationships between various dimensions of sustainability and associated resiliency. This perspective would also consider the different roles of resilience and sustainability in tourist destinations of different size, type and levels of infrastructure development. To this end, we have proposed a general system of equations to represent the resilience component of a dynamic model of sustainable tourism. While the model is simple and highly stylised, it is our hope that it will stimulate discussion of the specific quantities and qualities of destination quality that define the sustainability and resiliency problem.

Finally and perhaps most importantly, the model suggests research needs that are critical to improved and more comprehensive understanding of sustainable tourism. While the specific functional forms and assumptions of the illustrated model may be subject to revision and tailoring to specific destinations, the fundamental relationship among tourism, dimensions of sustainability, and the benefits of tourism realised by different stakeholder groups will almost certainly be central to any effective treatment of tourism sustainability — formal or informal. Among other needs, operational treatments of tourism sustainability will require appropriate measures of the short- and long-term benefits (and costs) of tourism, beyond operational measures of tourism development or visitation. Such measures extend beyond simple measures of economic activity commonly reported in the tourism literature. Also needed is an improved understanding of the dynamic relationships among various dimensions of sustainability, tourist activity and resulting short- and long-term tourism benefits. It is these latter dynamics that have been the principal focus here.

Such research needs are not trivial. Full and comprehensive application of dynamic optimisation models to tourism development will likely remain impractical for most tourist destinations, at least for the foreseeable future. Nonetheless, exploration of such mathematical models in a general, qualitative sense may provide tourism researchers and practitioners with improved means to conceptualise trade-offs implicit in tourism sustainability, and a means to incorporate a guiding structure to an area of debate often characterised by a lack of theoretical, conceptual and practical clarity. It is our hope that this paper has contributed, in a small way, to this goal.

Notes

¹ Johnston and Tyrrell (2005) detail the various assumptions implicit in equation (2). For example, this equation implies that each visitor uses up a constant m units of environmental quality per period. That is, V visitors will result in a loss of mV index-units of environmental quality per period. The simplest mathematical case for analysis is the case in which m=1, or in which each visitor causes 1 index-unit of degradation per period. Hence, for mathematical simplification, they scale the environmental quality index such that m=1. The result is that V visitors will cause exactly V index-units of environmental quality to be lost, per period. While this simplification may be easily relaxed, doing so does not alter fundamental aspects of the model.

² Although one could structure the model to reflect net benefits realised by any group affected by tourism.

³ In an urban setting, site attractiveness or 'environmental' quality may be related to such elements as the perceived authenticity of the cultural surroundings or the extent to which pedestrian areas are attractive and well kept. In contrast, the relevant definition of environmental quality in ecotourism destinations will most likely relate to the degree to which natural ecosystems are perceived as pristine and unspoiled.

⁴ It is also possible for planners to directly influence environmental quality (eg investing in urban infrastructure, site maintenance, and natural areas restoration).

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