Maintaining the productivity of co-culture systems in the face of environmental change

Co-culture systems can address food security issues by sustainably intensifying production of crops and animal protein without requiring additional land area. We show how a graph-theoretic optimization model based on ecological network analysis can determine robust co-culture strategies by controlling the presence of key species. Results of simulations on a hybrid rice and crayfish production system indicate that comparable levels of productivity can be achieved with different ecological network structures. Systems that combine agriculture and aquaculture have the potential to intensify food production relatively sustainably. This study shows how models of such co-culture systems based on ecological networks can optimize production based on the species involved.

disjoint sets of nodes to represent ecosystem components (O-type 'ecosystem functional unit' nodes represented by horizontal bars) and their functions (M-type 'ecosystem service' nodes represented by circles); this distinction allows a single species to be represented fully if it plays multiple roles in an ecosystem 13 . No two nodes within the same set of the graph are connected by an arc. Three component algorithms are then used to generate and optimize network structures (Methods). To illustrate the additional insights that can be drawn from our technique, we use the rice-crayfish co-culture farming system (Supplementary Fig. 1) modelled elsewhere 6 . That model was developed and validated using the state-of-the-art in Ecopath methodology considering both plausibility and data fit 15 .
The farm system consists of 13 different species interacting in a food web. The socio-ecological process graph is used to perform a combinatorically complete search of ecological interactions to generate candidate structurally feasible networks. The resulting structures can serve as a skeletal framework for subsequent detailed analysis that considers the magnitudes of the flows and interactions. The search algorithms use constraints that include mass balances and the existence of known ecological interactions 6 . The search domain for finding feasible ecological interactions in the ecosystem can be reduced by eliminating or replacing species without losing any ecological interactions. The details of this ENO approach are described in Methods.
The co-culture system simultaneously produces rice and crayfish using the same land area without an external supply of supplements. The diverse ecosystem embedded in this farm includes aquatic plants, phytoplankton, benthic animals and zooplankton. The purpose of the modelling is to identify robust optimal networks that maximize the system productivity per unit land area; if the co-culture system is designed to maintain the yield of the primary crop 16 , the productivity can also be measured in terms of the crayfish-rice ratio. We also assume that the crayfish has a flexible diet 17 . Selected interactions among functional groups or species are illustrated in the Supplementary Information. Simulations were performed to maximize crayfish production per unit of rice. Solving the model generates 60 alternative optimal and near-optimal solutions from a maximal structure that contains all possible network linkages (Supplementary Information). Differences in productivity result from varying distribution of nutrients within these structures. Figure 1 shows the structure that provides optimal harvest or production of 80.23 × 10 −2 kg of crayfish per kg of rice (Supplementary Table 1). The optimal harvest can be obtained with fixed/assumed supply of aquatic plants, rice, phytoplankton and detritus as food sources for crayfish (Fig. 1a,c). The network is much sparser than the maximal one, whose structure can be seen in the greyed-out parts of the figure (or see Supplementary Fig. 1 Figure 2 shows the trends with changing ecosystem structure against the number of species present and productivity based on the mass of crayfish per unit of rice, the latter being a proxy for land area. The key result is that there are families of ecosystem structures that differ in the number and type of biological species but have nearly identical productivity. For example, structures 1 to 4 have similar levels (∼0.12 kg of crayfish per kg of rice) but the number of species sequentially varies from five to nine. All 60 structures contain crayfish, plants, rice, phytoplankton and detritus, with structures 1 and 4 having only these five species. Structure 2 includes rotifera, bacteria and bacterioplankton and structure 3 additionally has cladocera. These are biologically different ecosystems with similar productivity. Note that there are several sequences or families of structures where productivity is robust and varies minimally if the species present significantly change. These 15 solution structures are shown in more detail in the Supplementary Information.
For the family of structures 5 to 15, the productivity varies from 0.095 kg kg −1 to 0.097 kg kg −1 , or about 20% lower than the optimal solution, while the number and type of species varies from four to ten. All the ecosystem structures include crayfish, plants, rice and phytoplankton. Different combinations of the other species are present in these solutions. The biological functions of the missing species are taken by the remaining species that occupy similar ecosystem niches. Note that structure 14 has the largest number of species and the most redundancy in this family.
Our results demonstrate how the analysis of alternative network structures can be used to ensure productive co-culture systems. Rather than identifying a unique (and possibly brittle) optimum, the process graph approach is used to identify families of solutions which translate into options that give good productivity while being robust to disruptions. These solutions provide guidance on the management of co-culture operations but, in practice, controlling species present will require further research work. Such information can be used to guide future agronomic co-culture field experiments by identifying potentially favourable system configurations (for example, ditch shape or surface area) and farming practices 16,18 . This approach has general applicability to a wide range of networks beyond the co-culture systems examined here, including portfolio optimization 19 of 'poly-culture' food production.

Methods
The socio-ecological process graph is a bipartite graph consisting of M-type nodes (circles) to represent ecosystem services and O-type nodes (horizontal bars) to represent ecosystem functional units. This modelling technique is intended for the analysis and optimization of ecosystems for provision of human needs 13 and is based on a class of engineering design problems known as process network synthesis (PNS) 20 . Arcs indicate interactions between ecosystem services and ecosystem functional units. If the M-type node is linked to more than one O-type node, it can be produced interchangeably by any combination of the latter. Inputs to an O-type node indicate ecosystem services necessary for its survival. Some ecosystem services can be classified as exogenous (originating from outside the ecosystem) or terminal (exiting the ecosystem for human use). The latter may be regarded as the final product of the ecosystem. Ecosystem services that are produced and consumed entirely within the ecosystem network are classified as intermediates. In the special case where only trophic linkages are considered, this technique uses assumptions that are compatible with those of static Ecopath models.
We have the following core assumptions 13,20 . SE1: All the terminal ecosystem services must be generated by the ecosystem solution structure. SE2: An ecosystem service represented in the structure is exogenous if, and only if, it is not an output of a functional unit defined in the ecosystem structure. SE3: Only those ecosystem functional units that are defined in the ecosystem can appear in an ecosystem solution structure. SE4: Any ecosystem functional unit has at least one path leading to a terminal ecosystem service. Every ecosystem functional unit contributes to the terminal ecosystem service. This assumption ensures that all functional units are non-redundant but are directly or indirectly involved in the generation of the final product. SE5: If an ecosystem service belongs to the ecosystem structure, it must be an input to or output from at least one ecosystem functional unit represented in the structure. In a socio-ecological process graph, the mass/energy is available to keep the ecosystem structure functioning and to meet the ecosystem services goal(s), such that a cost metric (for example, money or ecological footprint) is minimized.
The maximal structure generation (MSG) algorithm generates a network that is the union of all combinatorically feasible networks 20 . The solution structure generation (SSG) algorithm is capable of generating all combinatorically feasible networks, on the basis of enumeration of possible structures arising from relationships between ecosystem services and ecosystem functional units 13,20 . MSG and SSG focus on network connectivity and do not account for transfers between units. The accelerated branch-and-bound (ABB) algorithm is used to determine optimal and near-optimal structures by evaluation of the performance of candidate networks, while excluding infeasible and redundant networks 20 . Unlike the conventional branch-and-bound algorithm used to solve generic mixed integer linear programming models, ABB uses inherent information embedded in all PNS problems to reduce the size of the search space during optimization. For more details, the reader may refer to a recent book on process graphs 20 .
Reporting summary. Further information on research design is available in the Nature Research Reporting Summary linked to this article.

Software and code
Policy information about availability of computer code Data collection Simulations were done using the software P-graph Studio (Version 5.

2.2.2) developed by the Department of Computer Science and Systems
Technology at the University of Pannonia in Hungary . It is available free of charge for research purposes via www.p-graph.org.

Data analysis
No software used.
For manuscripts utilizing custom algorithms or software that are central to the research but not yet described in published literature, software must be made available to editors and reviewers. We strongly encourage code deposition in a community repository (e.g. GitHub). See the Nature Portfolio guidelines for submitting code & software for further information.

Data
Policy information about availability of data All manuscripts must include a data availability statement. This statement should provide the following information, where applicable: -Accession codes, unique identifiers, or web links for publicly available datasets -A description of any restrictions on data availability -For clinical datasets or third party data, please ensure that the statement adheres to our policy Data used in the analysis can be obtained via https://github.com/EcologicalP-Graph/Optimal-rice-crayfish-co-culture-system-/find/main.