A mathematical model of national-level food system sustainability
Conor Goold, Simone Pfuderer, William H. M. James, Nik Lomax, Fiona Smith, Lisa M. Collins
AA mathematical model of national-level food systemsustainability
Conor Goold , Simone Pfuderer , William H. M. James , Nik Lomax , FionaSmith , and Lisa M. Collins Faculty of Biological Sciences, University of Leeds, LS2 9JT, UK School of Agriculture, Policy and Development, University of Reading, Reading, RG6 6AR, UK School of Geography and Leeds Institute for Data Analytics, University of Leeds, LS2 9JT, UK School of Law, University of Leeds, LS2 9JT, UK
Abstract
The global food system faces various endogeneous and exogeneous, biotic and abi-otic risk factors, including a rising human population, higher population densities, pricevolatility and climate change. Quantitative models play an important role in under-standing food systems’ expected responses to shocks and stresses. Here, we present astylised mathematical model of a national-level food system that incorporates domesticsupply of a food commodity, international trade, consumer demand, and food commod-ity price. We derive a critical compound parameter signalling when domestic supplywill become unsustainable and the food system entirely dependent on imports, whichresults in higher commodity prices, lower consumer demand and lower inventory levels.Using Bayesian estimation, we apply the dynamic food systems model to infer the sus-tainability of the UK pork industry. We find that the UK pork industry is currentlysustainable but because the industry is dependent on imports to meet demand, a de-crease in self-sufficiency below 50% (current levels are 60-65%) would lead it close to thecritical boundary signalling its collapse. Our model provides a theoretical foundationfor future work to determine more complex causal drivers of food system vulnerability.
Food security is defined as “when all people at all times have physical, social and economicaccess to sufficient, safe and nutritous food to meet their dietary needs and preferencesfor an active and healthy life” [26]. The realisation of food security depends on the three1 a r X i v : . [ ec on . GN ] D ec REPRINT pillars of access, utilisation and availability [40, 6], and therefore is an outcome of coupledagricultural, ecological and sociological systems [30, 22, 33]. In recent years, the resilienceof food systems has become a priority area of research [46, 68, 7, 57] as biotic and abiotic,endogeneous and exogeneous demands on food systems grow, and the deleterious effects foodsystems currently have on the environment become more apparent [61, 65]. The challengeof meeting the requirements of food security is for food systems to expand their productioncapacities while remaining resilient to unpredictable perturbations and limiting their effectson the environment, such as reducing waste [21].Food systems research is inherently transdisciplinary [20, 30], of which one strand iscomputational and mathematical modelling. One utility of quantitative modelling is theability to build and perturb realistic ‘systems models’ of food systems to project importantoutcomes, such as future food production levels, farmer profitability, envirnomental degrada-tion, food waste, and consumer behaviour [61, 38, 54, 66, 55, 5]. The difficulty in modellingfood systems is their complexity, frequently resulting in large models with tens to hundredsof parameters and variables (e.g. [54, 61]), which are challenging to analyse and even morechallenging to statistically estimate from noisy real-world data [63]. In contrast, a handfulof authors have used relatively simple, theoretical models that are more amenable to formalanalysis, and have fewer parameters to estimate from data. For example, [66] link popula-tion dynamics to food availability and international trade using a generalised logistic model.[69] recently reported that the global food system is approaching a critical point signallingcollapse into an unsustainable regime by condensing the multi-dimensional global food tradenetwork into a bi-stable, one-dimensional system (using the framework of [27]). Simple mod-els of coupled ecological, economic and agricultural processes have also been investigated,such as to explain the emergence of poverty traps, which have direct effects on individual’saccess to food [44].While simplified models are less suitable for making predictions of complex systems’ out-comes, their tractability makes them better placed to elicit causal explanations and generatehypotheses of how systems work [59, 58, 48]. These stylised models are the backbones of sci-entific disciplines such as ecology [41], evolutionary biology [9], epidemiology [35], economics[43], and physics [64]. For instance, stylised mathematical models of brain networks, animalcollectives, ecosystems and cellular dynamics have been used to find the critical points atwhich they show abrupt qualitative changes in their behaviours [60, 56]. Food systems re-search, however, lacks such foundational models. Research on commodity production cycleshas developed models to couple agricultural production, supply chains, consumer demand,price and human decision-making [42, 63]. However, applications of systems dynamics modelsof commodity cycles are frequently high dimensional, encoding multiple modes of behaviourthat are not easily amenable to standard mathematical analysis. Gaining greater theoreti-cal insight into the dynamics of food systems and food security would be aided by simplermodels of coupled agri-food systems. 2REPRINT
In this paper, we develop and analyse a stylised model of a food system inspired by thesystems dynamics modelling of [42] and [63], yet simple enough to offer general theoreticalresults. We focus on modelling a national-level food system, where the effects of internationaltrade on domestic production is examined. Like the stylised models used to understand thecausal processes in evolution, epidemics, and ecological interactions, our approach necessarilyignores many important features of real food systems. Nonetheless, its relative simplicityallows us to elucidate the precise conditions under which different stable modes of behaviourimportant to food system resilience emerge in our system.We apply our theoretical model to the case of the UK pork industry, a key contributor tothe UK meat industry which currently employs 75,000 employees and is worth £1.25 billion[15]. Historically, pig industries have been of much interest to economists and agronomistsas one of the first investigations into business or ‘pork’ cycles [29, 12, 24, 31, 42, 71, 49, 63].Business cycles reflect the oscillations between commodity prices and supply, which havebeen posited to be the result of both endogeneous (e.g. [43]) and exogeneous mechanisms(e.g. see [28]). Over the last 20 years, however, the size of the UK pig industry has decreasedby approximately 50%, from 800,000 to approximately 400,000 sows, due to a combinationof legislative, epidemiological, and trade-related issues [67, 13]. Following the ban on sowgestation crates in 1999, as well as disease outbreaks in the early 2000s, imports of pig meatincreased by 50% [16], exceeding domestic production. While domestic production has re-turned to accounting for around 60-65% of total supply [16], the UK pig industry is still atrisk from high costs of production [23] and ‘opportunistic dealing’ within the pork supplychain favouring cheaper imports [8]. The sustainability of the UK pig industry is, thus, aconcern for UK food system resilience, particularly with the incipient threats of Brexit andthe COVID-19 pandemic that are affecting international trade, labour availability, commod-ity prices, and consumer demand [52, 25, 51]. We demonstrate how our food systems modelcan be used to infer the sustainability of the UK pork industry using Bayesian estimation,enabling us to quantify full uncertainty in parameter estimtates.
Our food system model is composed of coupled ordinary differential equations, with thestate variables of capital, inventory, consumer demand, and price (Figure 1; see variableand parameter definitions in Table 1). While we focus on food commodities, the model’sgenerality means that it could be applied to other types of commodity. Capital representsa raw material used to gauge the viability of the domestic industry, which could represent,for instance, the number of animals in the breeding herd for meat industries (e.g. [42])or the number of paddy fields in rice supply chains (e.g. [11]). Inventory is the stock of3REPRINT
C IDP a ( C, P, b ) e ( C ) f ( g, C ) w ( I ) k ( h, f gC ) c ( I, D ) m ( D, P, h ) r ( D, I, P ) Figure 1:
The general structure of the theoretical complex food system model. Blue circles denote the thefour state variables (capital, inventory, demand and price). Solid arrows indicate the different flows into andout of each state variable comprising their rate of change, and the arrow labels display generic functions ofthe model state variables and parameters (see Table 1 for the specific definitions used in this model). Dashedarrows show dependencies between different state variables and flows. processed food commodity being investigated. consumer demand represents the amount ofinventory demanded per time unit by the population of consumers, and is dependent on thecommodity price. The commodity price represents the price received by producers per unitof commodity produced, although we do not distinguish between producer and retail priceshere (i.e. the producer price is assumed to be directly proportional to the retail price). Whilemany of the mechanisms in supply chain functioning may be represented as a discrete-timesystem, we assume the aggregate behaviour of the national-level food system is adequatelyapproximated in continuous time by a system of differential equations.4REPRINT
Capital changes according to the equation: dCdt = aC (cid:16) Pb − (cid:17) − eC , C (0) = C (1)with initial condition at t = 0 , C . The parameter a is the rate of capital change (increaseor decrease) depending on the price to capital production cost ( b ) ratio. Captial depreciatesat rate e , where e − is the average life-time of capital.Inventory changes according to: dIdt = f gC − wI − IsD + I D + k ( h − f gC ) , I (0) = I (2)The first term represents the amount of inventory generated by domestic capital per timeunit (i.e. domestic supply), where f is a production rate, and g is a conversion factorrepresenting the amount of inventory units produced per unit of capital. Inventory is wasted(i.e. produced but not consumed) at rate w . The third term denotes the rate of inventoryconsumption by consumers, which is a non-linear Holling type-II/Michaelis-Menten functionasymptoting at I for D >> I . The dimensionless function I/ ( sD + I ) can be interpretedas the proportion of the demand that can be satisfied with the current inventory level.The parameter s is the ‘reference coverage’ converting inventory demanded per time unitinto commodity units, and is interpreted as the number of time units-worth of inventoryprocessors desire to have in stock. Perishable food commodities (e.g. meat) will have a lowerreference coverage, whereas less perishable items (e.g. rice, flour) can be stored for longerperiods and, therefore, stock levels can be controlled by increasing s , lowering the proportionof demanded units satisfied.The final term in equation 2 represents international trade, and its formulation cancommunicate different dynamics between domestic producers, processors and retailers. Weretain simplicity by assuming that trade is proportional to the difference between a referencedemand level ( h ) and current domestic production ( f gC ). When h > f gC , inventory is im-ported, and when h < f gC , inventory is exported. Realised demand ( D ) is a function of h and the current commodity price (see below), and therefore h represents the expected, base-line demand all else being equal [63]. Trade levels adapt to the reference demand to avoida positive feedback between higher prices, lower demand, and collapse of the commoditymarket for countries that are net importers, or a positive feedback between low prices, highdemand, and exponentially increasing production for net exporters. In some industries, in-cluding the UK pork industry, cheaper international imports lower the domestic commodityprice [1], and thus importing more than domestic supply when demand drops due to higherprices is a mechanism for lowering the commodity price and increasing demand. The differ-ence between reference demand and domestic production that is traded, however, is limitedby factors such as trade tariffs (e.g. [25]) or the ability of a nation to attract trade partners,and thus the parameter k controls the proportion of this difference. For countries that rely5REPRINT on international trade to supplement domestic supply to meet demand (net importers), − k represents the self-sufficiency of the domestic industry (i.e. the percentage of total suppliesproduced domestically).The instantaneous rate of change in demand is modelled as a simple function of referencedemand and the commodity price to reference price ratio: dDdt = m (cid:16) h qP − D (cid:17) , D (0) = D (3)The parameter m controls the time-responsiveness of demand. The reference price q istypically interpreted as the price of substitute items [63] or could represent consumers’ overallwillingness to pay. When the current price exceeds the reference price, demand falls, andvice versa.Many models exist for describing the price of commodities (e.g. see [37, 14] for someexamples), and we adopt a relatively simple formulation here that has the rate of change ofprice depend only on the coverage: dPdt = rP (cid:16) sDI − (cid:17) , P (0) = P (4)The coverage is a dimensionless quantity representing the amount of commodity needed tosustain current demand for s time periods divided by the current inventory level. At rategiven by r , the price increases when the coverage exceeds one, sD/I > , and decreases whencoverage falls below .To make our model more generalisable, we non-dimensionalise the system of equationsabove (see supplmentary materials for non-dimensionalisation) using the dimensionless quan-tities in Table 2, which reduces the number of parameters from 12 to 8. The non-dimensionalisedsystem of equations is: dvdτ = v (cid:16) αz − (cid:17) − βv (5) dxdτ = δv − ωx − γ xyy + x + κ ( γ − δv ) (6) dydτ = µ (cid:16) z − − y (cid:17) (7) dzdτ = ρz (cid:16) yx − (cid:17) (8)where { v, x, y, z } now represent the dimensionless state variables, τ is rescaled time, and thedimensionless parameter groups are denoted by Greek letters.6REPRINT Symbol Definition Units
Variables C Capital [ C ] I Inventory [ I ] D Demand [ It − ] P Price [
P I − ] t Time [ t ] Parameters a Capital growth rate [ t − ] b Cost of capital production [
P I − ] e Capital depreciation rate [ t − ] f Capital production rate [ t − ] g Capital conversion factor [ IC − ] w Inventory waste rate [ t − ] s Reference coverage [ t ] k Trade strength [ − ] h Reference demand [ It − ] m Demand response rate [ t − ] q Reference price [
P I − ] r Price growth rate [ t − ] Table 1:
Symbols, definitions and their units for the complex food system model.
A range of data is collected on the UK pork industry, but raw time series data is onlyavailable for certain variables and time frames. To fit our theoretical model, we focused on
Symbol Definition Description
Variables v CC Rescaled capital x Ihs
Rescaled inventory y Dh Rescaled demand z Pq Rescaled price τ t /a Rescaled time
Parameters α q/b
Reference profitability β e/a
Capital replacement-depreciation ratio δ f gC / ( ahs ) Initial production-demand ratio ω w/a
Waste-production ratio γ / ( as ) Capital replacement-coverage ratio κ k
Trade strength µ m/a
Demand response-capital replacement ratio ρ r/a
Price response-capital replacement ratio
Table 2:
Symbols and definitions for the dimensionless complex food system model.
Variable Data Details t Time set to monthly intervals between 2015through 2019 Price data only available for this timeperiod C Number of female pigs in the breeding herd(June and December surveys) [17] The breeding herd represents the maincapital of meat industries I Amount (kg) of new pork available for con-sumption [16, 3] Calculated as UK production (from[16]) plus imports and minus exports(from [3]) D No data available Demand is a latent quantity P All pig price (kg/deadweight) [18] The price producers receive, assumed tobe proportional to the retail price
Table 3:
UK pork industry data sources used to fit the food systems model monthly data over a period of 5 years from 2015 through 2019, which covers the availableannual data for the ‘All pig price’ per kilogram of deadweight (i.e. a combined price forstandard and premium pigs). All data sources used to fit the model are presented in Table3. Monthly data for the inventory of pork, taking into account current levels of consumptionand waste (e.g. the amount held in cold storage), is not reported in the UK. However, as anapproximation, we used the total new monthly supplies, calculated as domestic production ofpig meat plus imported pig meat minus exported pig meat. No data is available on consumerdemand, as this is a theoretical quantity. Missing data was considered missing completely atrandom (i.e. ignorable) because data collection schemes are largely independent and fixed.For instance, missing breeding herd data were not considered dependent on the price or newsupplies data.
The parameters and initial conditions of the non-dimensionalised model were estimated usingBayesian estimation in the probabilistic programming language Stan [10] using the RStaninterface in R [62, 53] using Stan’s Runge-Kutta 4th and 5th order integration scheme (seeStan code in the supplementary materials). The available monthly time series data, Y , formonth i and state variable j was assumed log-normal distributed (to ensure positivity): Y ji ∼ Lognormal ( ln ( Z j ) , σ j ) (9)where Z j is the state variable computed from the food systems model. In addition to fittingthe state variables of the model to the time series data, we fit the UK monthly productionfigures, and the monthly imports and exports, to the respective flows from the model:Production ∼ Lognormal ( ln ( f gZ ) , (cid:15) ) (10)8REPRINT Imports ∼ Lognormal ( ln ( kh ) , (cid:15) ) (11)Exports ∼ Lognormal ( ln ( kf gZ ) , (cid:15) ) (12)To aid computation, all parameters were transformed to a similar scale and given standardunit normal prior distributions (see full model specification in the supplementary materials),and were back-transformed to the appropropriate scale when integrating the model. We didnot estimate the parameters b (cost of capital production) and g (conversion factor fromcapital to inventory units) because these were known with enough certainty beforehand: b was set to the 138.3 p/kg (the average cost of production between 2015 and 2020), and g wasset to 82.4 kg/pig, reflecting the average slaughter weight of pigs (109.9kg) multiplied by a75% dressing yield (0.75) most recently reported by [2]. We ran 4 Markov chain Monte Carlo(MCMC) chains consisting of 2,500 iterations of warmup and 2,500 iterations of sampling,providing 10,000 samples from the posterior distribution for inference. All chains ran withoutany divergent transitions, and all parameters had effective sample sizes >> and ˆ R statistics (i.e. the Gelman-Rubin diagnositc) . < ˆ R < . indicating convergence.Each parameter is summarised by its mean and 95% highest density interval (HDI, the most95% most likely values). All data and code are available at https://github.com/cmgoold/cfs-model. No explicit solutions to the four-dimensional system of non-linear equations exist. Nonethe-less, its dynamics can be summarised by investigating its stable modes of behaviour. Toinvestigate stability, we conduct linear stability analyses [64]. Linear stability analysis isbased on a Taylor series expansion in multiple variables around the fixed points ( { ˆ v, ˆ x, ˆ y, ˆ z } ),where asymptotic (i.e. t → ∞ ) stability to small perturbations can be inferred when thereal part of the eigenvalues of the system’s matrix of partial derivatives (the Jacobian ma-trix representing the linearisation around the fixed points) evaluated at each equilibrium arenegative. Notably, the inverse of the leading eigenvalue of the Jacobian matrix determinesthe ‘characteristic return time’ of the system, with more resilient systems returning morequickly to their equilibria following a disturbance [50]. When international trade is not present (i.e. κ = 0 ), there is one stable fixed point ofthe system given by the state variable values (cid:8) α (2 ω + γ )2 δ (1+ β ) , α β , α β , βα (cid:9) . Another fixed point9REPRINT is where all state values are 0 (i.e. no industry). Conducting a linear stability analysisaround the latter fixed point, the eigenvalues of the Jacobian matrix at this equilibria are { ( − − β ) , − ω, − µ, − ρ ) } . All parameters are defined to be positive (except κ , which is here), meaning there are no conditions where the ‘no industry’ equilibria will be stable wheninternational trade is absent. In other words, the domestic industry is always viable. When < κ < , international trade is possible, opening the possibility of competition be-tween domestic and international products, or of an export market for domestic production.An unstable equilibria still exists where all state variables are 0, except for ˆ x = κγω becauseimporting products is possible. However, in addition, there is 1) an unsustainable domesticproduction equilibrium, where the food system is reliant on international imports, and 2)a sustainable domestic production equilibrium, where the domestic industry co-exists withinternational trade.The unsustainable domestic production equilibria is given by the set of fixed points { , κγω + γ , κγω + γ , ω + γ κγ } . The Jacbobian matrix ( J ) at this equilibria evalutes to: J (cid:12)(cid:12)(cid:12)(cid:8) , κγω + γ , κγω + γ , ω + γ κγ (cid:9) = α ( ω + γ ) κγ − − β δ (1 − κ ) − ω − γ − γ
00 0 − µ − µ ( κγω + γ ) − ρ ( ω + γ κγ ) ρ ( ω + γ κγ ) (13)and its eigenvalues ( λ ) are the roots of the fourth-degree characteristic polynomial: (cid:16) α ( ω + γ ) κγ − − β − λ (cid:17)(cid:104) − λ + (cid:0) − ω − γ − µ (cid:1) λ + (cid:0) − µ ( ω + ρ ) − γ (cid:1) λ − γ µρ (cid:105) = 0 (14)The first eigenvalue can be determined directly as: λ = α ( ω + γ ) κγ − − β (15)By using the Routh-Hurwitz conditions, the sign of the remaning eigenvalues’ real parts [47]will always be negative (see the supplementary materials). Ultimately, the unsustainabledomestic production mode will be stable if:critical ratio = α ( ω + γ ) κγ (1 + β ) < (16)The dependence of this critical ratio to changes in the original parameter values is shown inFigure 2a. The numerator represents a weighting of three factors: i ) the reference profitability of capital production ( α ; Table 2), ii ) the need for new commodity , where either higher10REPRINT rates of waste or greater reference coverage increases the critical ratio, and ii ) the speed ofcapital production , where higher rates ( a in Table 1) increases the critical ratio (Figure 2a).By contrast, the denominator represents the total strength of international trade, which iscomposed of the trade strength parameter ( κ ), weighted by the viability of domestic capital:if the capital production rate increases, or capital depreciation rate decreases, the total tradestrength becomes smaller, serving to increase the critical ratio.When the critical ratio exceeds 1, the sustainable domestic production equilibrium isgiven by { γκ ( − − β )+ α ( γ +2 ω )2 δ (1+ β )(1 − κ ) , α β , α β , βα } . In the latter case, the equilibrium values of in-ventory, demand and price are the same as the equilibrium values of the model withoutinternational trade (see above), determined by the profitability of the domestic industry andthe ratio of capital depreciation and growth rates. However, equilibrium inventory and de-mand are lower, and equilibrium price is higher, than when domestic supply is unsustainable.For example, the conditions for equilibrium price when domestic supply is unsustainable tobe higher than when domestic supply is sustainable (i.e. ω + γ κγ > βα ) is exactly the criticalratio. The latter trends are also seen the closer a system comes to the critical ratio. Thus,while international trade in the short term increases inventory levels, decreases the coverage(see model description) and, therefore, decreases the price and increases demand, the long-term result of the unsustainable domestic production regime is higher prices, lower demandand lower inventory levels.The equilibrium value of capital is similar to the equilibrium value found in the modelwithout international trade, but now factors in trade strength, κ (both equilibria are equalwhen κ = 0 ). Specifically, whether κ has a positive or negative influence on the long-termsustainable equilibrium domestic capital depends on whether the system is characterised bynet imports (domestic supply is less than reference demand) or net exports (domestic supplyexceeds reference demand), i.e. whether γδ − ˆ v in equation 6 is greater than or less than zero.If domestic supply is less than reference demand, and the critical ratio exceeds 1, increasingtrade strength will reduce the equilibrium value of capital in the long term limit (i.e. as τ → ∞ ). However, if domestic supply exceeds reference demand, increasing trade strengthwill increase the equilibrium capital due to a greater ability to export surplus product. Fromthe previous inequality, we can define the surplus ratio, which signals that domestic supplywill be greater than reference demand (net exports) if:surplus ratio = α ( ω + γ ) γ (1 + β ) ≡ κ · critical ratio > (17)which equals the critical ratio cancelling out the trade strength. Figures 2b-d demonstratethe relationship between the sustainable and unsustainable stable modes of behaviour in ( κ, α ) space for differing values of β , as well as the distinction between the sustainable statecharacterised by net imports or exports. 11REPRINT Figure 2:
Stability of the model incorporating international trade ( < κ < ). Panel a shows the sensitivityof the critical ratio to parameters in Table 1. The x-axis shows the ratio of the parameters to their referencevalues ( q = 160 , b = 140 , e = 0 . , a = 0 . , w = 0 . , s = 1 , k = 0 . ). The horizontal dashed line shows thecritical ratio threshold of unity. Panels b-d show the stable modes of behaviour in ( κ, α ) space for differingvalues of β , distinguishing between unsustianable (red), sustainable with imports (blue), and sustainable withexports (green) behaviours. Panels b-d are produced with the remaining parameters at γ = 26 , ω = 10 and δ = 5 . The critical ratio for the UK industry was estimated to be credibly above 1 (Table 4),suggesting the industry is in a sustainable condition according to this model. The referencedemand is estimated to be approximately 1.6 times that of the UK estimated pig meat annualconsumption (approximately 140 million kg based off 25 kg/person/year and a population12REPRINT
Parameter Mean 95% HDI ESS a e f k h w m q r s Table 4:
Key parameter estimates (mean and 95% highest density interval, HDI) and effective sample sizes(ESS) from fitting the model to the UK pig industry data. Parameters b and g were fixed at the constants138.3 p/kg and 82.4 kg/pig, respectively. size of 66.65 million people; [2]). The trade strength is approximately 0.36 on average, whichis consistent with the current self-sufficiency level of around 65% (i.e. around 35% of UKpig meat is imported). The difference between α and the surplus ratio is credibly less thanzero (mean: -0.64; HDI: [-0.83, -0.43]), reflecting that UK production of pig meat does notmeet expected demand. The critical κ value needed to push the UK domestic industry intothe unsustainable regime is 0.61 (95% HDI: [0.56, 0.65]).Posterior predictions (Figure 3) reflect the most plausible trajectories of the food systemmodel generating the data, and posterior predictive distributions (open blue circles) covera large proportion of the observed data. However, there are additional sources of variationthat the model trajectories do not account for. For instance, there is seasonal variation inUK pork production: the breeding herd tends to be higher in the July than in the Decembercensuses (Figure 3a), resulting in higher UK production of pig meat (Figure 3e) in thelatter portions of the year, likely in preparation for the Christmas period. The pig priceis also more variable than the model can explain (Figure 3d), showing a notable drop in2016 (corresponding to a fall in the EU pig price) and an increase in 2019. The differencebetween imports and exports (Figure 3f) fluctuated over the 2015-2019 time period, whereasthe model only considers a simple international trade function (equation 2). This paper has presented a theoretical model of a complex food system that balances an-alytical tractability and realism. The model represents the functioning of a national foodsystem including international trade, and we have shown that the sustainability of the do-mestic industry depends on a critical compound parameter that comprises the profitability13REPRINT
Figure 3:
Fitting the food systems model to the UK pig industry data. Orange filled circles show theraw monthly data (some data is missing), thin black lines display 200 random samples from the posteriordistribution, the thick black lines indicate the mean posterior trajectory, and open blue circles display 200random samples from the posterior predictive distribution (i.e. predictions incorporating random noise ). of the domestic industry (the reference price to cost of capital production ratio), the needfor new commodity (commodity waste rates and reference coverage), the ability to producenew capital (captial growth and depreciation rates), and the strength of international trade(see Figure 2a). Below unity, this critical ratio signals that international trade outcompetesthe domestic industry and the model enters an unsustainable domestic supply regime char-acterised by complete reliance on imports. This unsustainable regime also results in higherequilibrium commodity prices, lower inventory and lower consumer demand than when do-mestic supply is sustainable. By estimating the key parameters of this ratio from data onreal food systems, the sustainability of domestic industries, conditional on the assumptionsof the model, can be evaluated.Within the sustainable regime of the model, a key factor determining the long-termbehaviour of the food system depends on whether it is characterised by net imports ornet exports. In the context of the mathematical model, this is represented by whetherthe nation produces enough food to meet the reference demand ( h in Table 1). We findthat a food system that must supplement domestic supply with imports in order to meetdemand (a net importer) is more vulnerable to collapse because increasing trade potential(higher k values) results in reduced self-sufficiency. Food systems that produce a surplus ofdomestic commodity can benefit from increased trade potential by exporting more (Figure2), assuming that export markets are always available. This supports the current literatureon the importance of diversifying food commodity sources to ensure food system resilience.Rapid globalisation has meant that 23% of the food produced globally, and 26% of globalcalorie production, is traded [19, 69, 51], and the majority of the world’s diet is partlydependent on food imports [36] with several countries not producing enough food to satisfybasic, per-capita caloric intake [19]. While global food trade has led to, and encouragesfurther, diversification of diets, it has also led to a less resilient global food system. Thisis because most countries in the trade network rely on imports from a smaller number ofdominant, trade partners [36], results supported also by theoretical modelling [69]. Themodel here provides a critical boundary where a nation’s food system may become entirelydependent on imports, leading to higher commodity prices, lower consumer demand andlower overall inventory levels. By the same token, complete self-sufficiency is an unrealisticand potentially harmful goal where the food system is again only reliant on a single supplychain [32].Application of our model to 2015-2019 data from the UK pork industry demonstratedthat the industry is in the sustainable model regime, with the critical ratio estimated crediblyabove 1 (Table 4). While the UK pork industry has diminished in size over the last 20 years(by around 50% since the late 1990s) its current level of self-sufficiency is around 60-65%and its export market continues to grow due to lower production levels of the Chinese porkindustry (e.g. [4]). The results from our model support this state of the industry: the ‘tradestrength’ parameter ( κ ), which represents the proportion of the difference between reference15REPRINT demand levels and domestic production that can be traded, was estimated between 35 and37%. Crucially, the critical value of κ that would tip the UK industry into collaspe isestimated to be 61%, with a 95% highest density interval between 56 and 65%, suggesting ifself-sufficiency drops below 50%, the industry will be closely approaching unsustainability.The UK food system faces a number of challenges in the coming years, including the impactof no-deal Brexit on trade tariffs allowing retailers easier access to cheaper imported meat[25], the continuing COVID-19 pandemic on the production and processing efficiency of foodcommodities [52], increasing popularity of reduced meat, vegetarian, vegan and plant-baseddiets on UK meat demand [34], and the potential for an African Swine Fever epidemic in thepig herd [45, 39]. While under some scenarios, such as a bespoke Brexit trade deal with theEU, the price of pig meat might increase [25], representing a boost to domestic producers,our model indicates that this result is also consistent with a food system coming closer tothe critical boundary delineating collapse (i.e. there is a negative relationship between longterm price and the critical ratio) and, thus, might not be an advantage to the food systemas a whole.The mechanisms encoded in the model presented here are simple relative to the multi-factorial functioning of real food systems [22, 33]. For instance, we have assumed thatcommodity prices respond only to changes in the supply-demand balance and not extenalfactors such as global commodity prices and costs of production. Moreover, the model as-sumes that international trade responds only to the difference between reference demand anddomestic production, and that exports of domestic produce are always available, ignoringhow government regulations of trade flows may disrupt this scenario (e.g. pigmeat from pigsfed with feed additives such as ractopomine might fail government import regulations). Ourspecification that domestic capital changes proportional to the commodity price–productioncost ratio [63] ignores heterogeneity in the structure of food supply chains. While similarassumptions have been used to build more complex system dynamics models (e.g. [42, 63]),there is great scope for expanding our system of differential equations to reveal the impactof, for instance, dis-aggregated actors of the supply chain (e.g. breeding pigs versus slaughterpigs, producers versus processors), the causal effects of non-finanical drivers (e.g. preserv-ing pig health and welfare) on supply chain functioning, heterogeneity in the production,demand and trade of different product types (e.g. fresh pork versus bacon and sausages),or how external factors, such as the climate, will influence production and demand [70].Nonetheless, our model provides a description of a single food system supported by bothpast theoretical and empirical work, and thus offers key insights into the conditions thatpromote sustainability of domestic food industries.16REPRINT REFERENCES
Acknowledgments
This research is a part of the PigSustain project which is funded through the Global FoodSecurity’s ‘Resilience of the UK Food System Programme’, with support from BBSRC,ESRC, NERC and Scottish Government (grant number BB/N020790/1). We thank theDepartment for Environment, Food and Rural Affairs, and the Agriculture and HorticultureDevelopment Board for making the data used in this article freely accessible.
Author contributions
CG conceptualised the paper, developed and analysed the model, wrote the computer code,analysed the data, and wrote the first draft of the manuscript; SP reviewed model develop-ment and contributed to writing and reviewing the manuscript; WHMJ reviewed and con-tributed to writing the manuscript; NL reviewed and contributed to writing the manuscript;FS reviewed and contributed to writing the manuscript; LMC attained funding, helped con-ceptualise the model and paper, and reviewed and contributed to writing the manuscript.
Data and code accessibility
All data and code to reproduce the results of this article are available at https://github.com/cmgoold/cfs-model.
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